The value of C in (0, 2) satisfying the mean value theorem for the function f(x) = x(x-1)², x∈[0,2] is equal to
🥳 Wohoo! Correct answer
MVT: f'(c)=[f(b)-f(a)]/(b-a)
Calculate f(2)=2 and f(0)=0
Apply MVT: [f(2)-f(0)]/2 = f'(c)
Find f'(x)=2x(x-1)+(x-1)² and solve f'(c)=1
Forgetting to verify if c lies in (0,2)
😢 Uh oh! Incorrect answer, Try again
Use MVT formula and verify c lies in interval
MVT: f'(c)=[f(b)-f(a)]/(b-a)
Calculate f(2)=2 and f(0)=0
Apply MVT: [f(2)-f(0)]/2 = f'(c)
Find f'(x)=2x(x-1)+(x-1)² and solve f'(c)=1
Forgetting to verify if c lies in (0,2)
If [x] is greatest integer function not greater than x then ∫₀⁸[x]dx equals
🥳 Wohoo! Correct answer
[x] = n for n ≤ x < n+1
Break integral at integer points
Sum areas of rectangles
Use formula for sum of integers
Not breaking at integers
😢 Uh oh! Incorrect answer, Try again
Consider function behavior at integers
[x] = n for n ≤ x < n+1
Break integral at integer points
Sum areas of rectangles
Use formula for sum of integers
Not breaking at integers
The integral from 0 to π/2 of ∫√sin θ cos³θ dθ equals
🥳 Wohoo! Correct answer
du = -sin θ dθ
Use substitution u = cos θ
Change limits accordingly
Integrate polynomial in u
Not changing limits correctly
😢 Uh oh! Incorrect answer, Try again
Consider substitution technique
du = -sin θ dθ
Use substitution u = cos θ
Change limits accordingly
Integrate polynomial in u
Not changing limits correctly
If eʸ + xy = e, the ordered pair (dy/dx, d²y/dx²) at x = 0 equals
🥳 Wohoo! Correct answer
d/dx(eʸ) = eʸ·dy/dx
Implicitly differentiate once
Implicitly differentiate twice
Substitute x = 0
Not differentiating implicitly
😢 Uh oh! Incorrect answer, Try again
Use implicit differentiation
d/dx(eʸ) = eʸ·dy/dx
Implicitly differentiate once
Implicitly differentiate twice
Substitute x = 0
Not differentiating implicitly
Evaluate ∫₂³ x²dx as limit of a sum
🥳 Wohoo! Correct answer
Limit of sum gives definite integral
Divide interval into n parts
Form Riemann sum
Take limit as n→∞
Not forming correct Riemann sum
😢 Uh oh! Incorrect answer, Try again
Use Riemann sum definition
Limit of sum gives definite integral
Divide interval into n parts
Form Riemann sum
Take limit as n→∞
Not forming correct Riemann sum
Integrate from 0 to π/2 ∫(cosx sinx)/(1+sinx) dx equals
🥳 Wohoo! Correct answer
du = cosx dx
Use substitution u = sinx
Change limits appropriately
Integrate rational function
Not changing limits correctly
😢 Uh oh! Incorrect answer, Try again
Look for substitution pattern
du = cosx dx
Use substitution u = sinx
Change limits appropriately
Integrate rational function
Not changing limits correctly
∫(cos 2x - cos 2α)/(cosx - cosα) dx equals
🥳 Wohoo! Correct answer
∫cos(ax) dx = (1/a)sin(ax)
Split into partial fractions
Integrate each term
Combine results
Not splitting correctly
😢 Uh oh! Incorrect answer, Try again
Look for pattern in numerator
∫cos(ax) dx = (1/a)sin(ax)
Split into partial fractions
Integrate each term
Combine results
Not splitting correctly
∫₀¹ xeˣ/(2+x)³ dx equals
🥳 Wohoo! Correct answer
∫udv = uv - ∫vdu
Use integration by parts
Handle cubic denominator
Evaluate at limits
Not handling denominator properly
😢 Uh oh! Incorrect answer, Try again
Consider parts carefully
∫udv = uv - ∫vdu
Use integration by parts
Handle cubic denominator
Evaluate at limits
Not handling denominator properly
Then sum of the degree and order of the differential equation (1 + y₁²)²/³ = y₂ is
🥳 Wohoo! Correct answer
Order = highest derivative, Degree = highest power after rationalization
Identify highest order derivative (y₂) = 2
Find degree by expressing in standard form
Add order(2) + degree(3) = 5
Not rationalizing properly
😢 Uh oh! Incorrect answer, Try again
Order is highest derivative, degree is highest power
Order = highest derivative, Degree = highest power after rationalization
Identify highest order derivative (y₂) = 2
Find degree by expressing in standard form
Add order(2) + degree(3) = 5
Not rationalizing properly
If dy/dx + y/x = x² then 2y(2) - y(1)
🥳 Wohoo! Correct answer
First order linear DE solution method
Solve DE to get y=x⁴/4 + C
Use y(1) to find C
Calculate 2y(2)-y(1) = 15/4
Not finding particular solution
😢 Uh oh! Incorrect answer, Try again
Solve for general solution first
First order linear DE solution method
Solve DE to get y=x⁴/4 + C
Use y(1) to find C
Calculate 2y(2)-y(1) = 15/4
Not finding particular solution
The solution of differential equation dy/(x+y) = 2dx is
🥳 Wohoo! Correct answer
Separable DE method
Let z = x+y to substitute
Separate variables
Integrate both sides
Not separating variables correctly
😢 Uh oh! Incorrect answer, Try again
Look for substitution pattern
Separable DE method
Let z = x+y to substitute
Separate variables
Integrate both sides
Not separating variables correctly
If y(x) be solution of Differential Equation xlog x·dy/dx + y = 2x log x, y(e) equals
🥳 Wohoo! Correct answer
Linear DE: dy/dx + P(x)y = Q(x)
Multiply by integrating factor
Solve resulting equation
Use initial condition y(e)
Not using integrating factor
😢 Uh oh! Incorrect answer, Try again
Find integrating factor first
Linear DE: dy/dx + P(x)y = Q(x)
Multiply by integrating factor
Solve resulting equation
Use initial condition y(e)
Not using integrating factor
If the parametric equation of curve is given by x=cosθ+log(tan(θ/2)) and y=sinθ, then the points for which dy/dx=0 are given by
🥳 Wohoo! Correct answer
dy/dx = (dy/dθ)/(dx/dθ)
Find dx/dθ and dy/dθ
Form dy/dx using chain rule
Set dy/dx=0 and solve
Not handling parametric equations correctly
😢 Uh oh! Incorrect answer, Try again
Use parametric differentiation
dy/dx = (dy/dθ)/(dx/dθ)
Find dx/dθ and dy/dθ
Form dy/dx using chain rule
Set dy/dx=0 and solve
Not handling parametric equations correctly
A particle starts from rest and its angular displacement (in radians) is given by θ=t²/20+t/5. If the angular velocity at the end of t=4 is k, then the value of 5k is
🥳 Wohoo! Correct answer
ω=dθ/dt
Find angular velocity by differentiating θ
Substitute t=4
Multiply result by 5
Not identifying angular velocity formula
😢 Uh oh! Incorrect answer, Try again
Angular velocity is dθ/dt
ω=dθ/dt
Find angular velocity by differentiating θ
Substitute t=4
Multiply result by 5
Not identifying angular velocity formula
∫(x³sin(tan⁻¹x⁴))/(1+x⁸) dx is equal to
🥳 Wohoo! Correct answer
d/dx(tan⁻¹x)=1/(1+x²)
Let u=tan⁻¹x⁴
Express integral in terms of u
Use substitution method
Not recognizing substitution pattern
😢 Uh oh! Incorrect answer, Try again
Look for composite function pattern
d/dx(tan⁻¹x)=1/(1+x²)
Let u=tan⁻¹x⁴
Express integral in terms of u
Use substitution method
Not recognizing substitution pattern
The value of ∫(xe^x)/(1+x)² dx is equal to
🥳 Wohoo! Correct answer
d/dx(e^x/(1+x))=(xe^x)/(1+x)²
Use integration by parts
Simplify numerator and denominator
Recognize standard form
Not recognizing reverse chain rule
😢 Uh oh! Incorrect answer, Try again
Consider derivative of product
d/dx(e^x/(1+x))=(xe^x)/(1+x)²
Use integration by parts
Simplify numerator and denominator
Recognize standard form
Not recognizing reverse chain rule
The value of ∫e^x[(1+sinx)/(1+cosx)] dx is equal to
🥳 Wohoo! Correct answer
Integrate e^x with trig functions
Rationalize numerator and denominator
Use trigonometric identities
Integrate term by term
Not using correct trig identities
😢 Uh oh! Incorrect answer, Try again
Look for combination of functions
Integrate e^x with trig functions
Rationalize numerator and denominator
Use trigonometric identities
Integrate term by term
Not using correct trig identities
If In = ∫[0 to π/4] tan^n(x) dx, where n is a positive integer.Find the value of I₁₀ + I₈
🥳 Wohoo! Correct answer
In = tan^(n-1)x/(n-1) - I(n-2)
Use reduction formula for tanⁿx
Apply formula for both I₁₀ and I₈
Add the results
Not applying reduction formula correctly
😢 Uh oh! Incorrect answer, Try again
Remember reduction formula for powers of tan
In = tan^(n-1)x/(n-1) - I(n-2)
Use reduction formula for tanⁿx
Apply formula for both I₁₀ and I₈
Add the results
Not applying reduction formula correctly
The value of ∫(√x) / (√x+√4042-x) dx from 0 to 4042 is equal to
🥳 Wohoo! Correct answer
Use symmetry to simplify
Use substitution u=√x
Transform limits accordingly
Apply integration formula
Not recognizing symmetry in limits
😢 Uh oh! Incorrect answer, Try again
Consider symmetric nature of integral
Use symmetry to simplify
Use substitution u=√x
Transform limits accordingly
Apply integration formula
Not recognizing symmetry in limits
The area of the region bounded by y=√(16-x²) and x-axis is
🥳 Wohoo! Correct answer
A = ∫ydx between limits
Recognize this is semicircle
Set up definite integral
Evaluate with proper limits
Not identifying curve shape
😢 Uh oh! Incorrect answer, Try again
Area under curve formula
A = ∫ydx between limits
Recognize this is semicircle
Set up definite integral
Evaluate with proper limits
Not identifying curve shape
Solution of Differential Equation xdy - ydx = 0 represents
🥳 Wohoo! Correct answer
y = mx is general solution
Separate variables
Integrate both sides
Interpret geometric meaning
Not interpreting geometric meaning
😢 Uh oh! Incorrect answer, Try again
Write as dy/dx = y/x
y = mx is general solution
Separate variables
Integrate both sides
Interpret geometric meaning
Not interpreting geometric meaning
The number of solutions of dy/dx = (y+1)/(x-1) when y(1)=2 is
🥳 Wohoo! Correct answer
Existence and uniqueness theorem
Check if equation is exact
Check for existence and uniqueness
Apply initial condition
Not checking uniqueness conditions
😢 Uh oh! Incorrect answer, Try again
Consider IVP conditions
Existence and uniqueness theorem
Check if equation is exact
Check for existence and uniqueness
Apply initial condition
Not checking uniqueness conditions
Value of ∫(1+x⁴)/(1+x⁶) dx is
🥳 Wohoo! Correct answer
• ∫dx/(1+x²) = tan⁻¹x + C
Split integral into partial fractions
Use substitution u = x³
Integrate to get arctangent terms
Not splitting fraction correctly
😢 Uh oh! Incorrect answer, Try again
Look for substitution patterns
• ∫dx/(1+x²) = tan⁻¹x + C
Split integral into partial fractions
Use substitution u = x³
Integrate to get arctangent terms
Not splitting fraction correctly
Value of ∫eˢⁱⁿˣ sin 2x dx is
🥳 Wohoo! Correct answer
• Integration by substitution • Chain rule
Let u = sin x, du = cos x dx
Use substitution method
Integrate and simplify
Not identifying correct substitution
😢 Uh oh! Incorrect answer, Try again
Look for relationship between integrand terms
• Integration by substitution • Chain rule
Let u = sin x, du = cos x dx
Use substitution method
Integrate and simplify
Not identifying correct substitution
The value of ∫cos⁻¹x dx integrated from -1/2 to 1/2 is
🥳 Wohoo! Correct answer
• ∫cos⁻¹x dx = x cos⁻¹x - √(1-x²) + C
Break integral into two parts at x=0
Use substitution u=cos⁻¹x
Evaluate at limits carefully
Not handling negative values correctly
😢 Uh oh! Incorrect answer, Try again
Remember cos⁻¹(-x) = π - cos⁻¹x
• ∫cos⁻¹x dx = x cos⁻¹x - √(1-x²) + C
Break integral into two parts at x=0
Use substitution u=cos⁻¹x
Evaluate at limits carefully
Not handling negative values correctly
The value of ∫ (log(1+x)) / (1+x²) dx integrated from 0 to 1 is
🥳 Wohoo! Correct answer
• ∫log(1+x) dx = xlog(1+x) - x + C
Use substitution x=tan θ
Transform integral limits
Use properties of logarithms
Not transforming limits correctly
😢 Uh oh! Incorrect answer, Try again
Consider relationship with arctangent
• ∫log(1+x) dx = xlog(1+x) - x + C
Use substitution x=tan θ
Transform integral limits
Use properties of logarithms
Not transforming limits correctly
The value of ∫ (cos x) / (1+eˣ) dx integrated from -π/2 to π/2 is
🥳 Wohoo! Correct answer
• Integration by parts • Symmetry
Use symmetry properties
Split integral at x=0
Combine results
Not using symmetry effectively
😢 Uh oh! Incorrect answer, Try again
Consider behavior of even/odd functions
• Integration by parts • Symmetry
Use symmetry properties
Split integral at x=0
Combine results
Not using symmetry effectively
Order of differential equation obtained by eliminating arbitrary constants in the family of curves c₁y = (c₂+c₃)e^(x+c₄) is
🥳 Wohoo! Correct answer
• Order of differential equation
Differentiate to eliminate constants
Count highest order derivative
Verify order
Counting order incorrectly
😢 Uh oh! Incorrect answer, Try again
Look for pattern in derivatives
• Order of differential equation
Differentiate to eliminate constants
Count highest order derivative
Verify order
Counting order incorrectly
General solution of differential equation x²dy-2xydx = x⁴cosx dx is
🥳 Wohoo! Correct answer
• I.F. method • General solution
Find integrating factor
Multiply throughout by I.F.
Solve resulting equation
Not identifying correct I.F.
😢 Uh oh! Incorrect answer, Try again
Look for integrating factor pattern
• I.F. method • General solution
Find integrating factor
Multiply throughout by I.F.
Solve resulting equation
Not identifying correct I.F.
Order of differential equation y = C₁e^(C₃+x) + C₃e^(C₄+x) is
🥳 Wohoo! Correct answer
Order = highest derivative
Count highest derivative needed
Check independent solutions
Determine minimum order needed
Counting constants instead of order
😢 Uh oh! Incorrect answer, Try again
Look at highest power of operator
Order = highest derivative
Count highest derivative needed
Check independent solutions
Determine minimum order needed
Counting constants instead of order
∫ 1 / (√x + x√x) dx =
🥳 Wohoo! Correct answer
Integration by substitution rules
Let u=√x to simplify
Use partial fractions if needed
Integrate resulting expression
Not making effective substitution
😢 Uh oh! Incorrect answer, Try again
Look for substitution to simplify
Integration by substitution rules
Let u=√x to simplify
Use partial fractions if needed
Integrate resulting expression
Not making effective substitution
∫₀¹√(1+x)/√(1-x) dx =
🥳 Wohoo! Correct answer
Integration by substitution formulas
Use substitution to simplify
Convert to standard form
Apply integration formula
Not making appropriate substitution
😢 Uh oh! Incorrect answer, Try again
Consider trigonometric substitution
Integration by substitution formulas
Use substitution to simplify
Convert to standard form
Apply integration formula
Not making appropriate substitution
∫x³sin3x dx =
🥳 Wohoo! Correct answer
∫udv = uv - ∫vdu
Use integration by parts
Apply formula multiple times
Collect like terms
Not handling signs correctly in IBP
😢 Uh oh! Incorrect answer, Try again
Consider repeated integration by parts
∫udv = uv - ∫vdu
Use integration by parts
Apply formula multiple times
Collect like terms
Not handling signs correctly in IBP
Integrating factor of differential equation (2x+3y²) dy = ydx (y>0) is
🥳 Wohoo! Correct answer
μ(x,y) makes equation exact
Identify standard form
Find integrating factor μ
Verify solution
Not recognizing standard form
😢 Uh oh! Incorrect answer, Try again
Check if equation becomes exact
μ(x,y) makes equation exact
Identify standard form
Find integrating factor μ
Verify solution
Not recognizing standard form
Equation of curve passing through (1,1) such that slope of tangent at any point (x,y) equals product of its co-ordinates is
🥳 Wohoo! Correct answer
dy/dx = xy given
Form differential equation
Use given condition
Solve with initial condition
Not using initial condition correctly
😢 Uh oh! Incorrect answer, Try again
Consider slope condition carefully
dy/dx = xy given
Form differential equation
Use given condition
Solve with initial condition
Not using initial condition correctly
∫(1 / [1+e⁻ˣ]) dx =
🥳 Wohoo! Correct answer
Integration by substitution
Substitute u = e⁻ˣ
Rewrite as ∫(e^x/(1+e^x)) dx
Integrate to get log(eˣ/(eˣ+1)) + c
Substitution errors
😢 Uh oh! Incorrect answer, Try again
Look for substitution
Integration by substitution
Substitute u = e⁻ˣ
Rewrite as ∫(e^x/(1+e^x)) dx
Integrate to get log(eˣ/(eˣ+1)) + c
Substitution errors
∫1 / (√3-6x+9x²) dx =
🥳 Wohoo! Correct answer
Integration of inverse trig
Complete square under root
Use substitution to get standard form
Integrate to get (1/3)sin⁻¹[(3x+1)/2] + c
Completing square errors
😢 Uh oh! Incorrect answer, Try again
Look for perfect square
Integration of inverse trig
Complete square under root
Use substitution to get standard form
Integrate to get (1/3)sin⁻¹[(3x+1)/2] + c
Completing square errors
∫eˢⁱⁿˣ ([sinx+1] / secx) dx =
🥳 Wohoo! Correct answer
Integration by substitution
Substitute u = sinx
Rewrite in terms of u
Integrate to get sinx·eˢⁱⁿˣ + c
Substitution errors
😢 Uh oh! Incorrect answer, Try again
Look for substitution
Integration by substitution
Substitute u = sinx
Rewrite in terms of u
Integrate to get sinx·eˢⁱⁿˣ + c
Substitution errors
∫₀¹ 1 / (eˣ + e⁻ˣ) dx =
🥳 Wohoo! Correct answer
Integration by substitution
Let eˣ = t to transform integral
Use partial fractions
Integrate to get tan⁻¹e - π/4
Substitution errors
😢 Uh oh! Incorrect answer, Try again
Consider substitution method
Integration by substitution
Let eˣ = t to transform integral
Use partial fractions
Integrate to get tan⁻¹e - π/4
Substitution errors
∫₀¹/² 1 / [(1+x²)√1-x²] dx =
🥳 Wohoo! Correct answer
Integration methods
Substitute x = sinθ
Convert to trigonometric form
Integrate to get (1/√2)tan⁻¹(√(2/3))
Substitution errors
😢 Uh oh! Incorrect answer, Try again
Trigonometric substitution
Integration methods
Substitute x = sinθ
Convert to trigonometric form
Integrate to get (1/√2)tan⁻¹(√(2/3))
Substitution errors
Area bounded by y = cosx between x = 0 and x = π is
🥳 Wohoo! Correct answer
Area = ∫|f(x)|dx
Divide into intervals where cosx changes sign
Find absolute value of integral
Get area = 2 sq units
Sign change errors
😢 Uh oh! Incorrect answer, Try again
Consider sign changes
Area = ∫|f(x)|dx
Divide into intervals where cosx changes sign
Find absolute value of integral
Get area = 2 sq units
Sign change errors
Area bounded by y = x, x-axis and ordinates x = -1, x = 2 is
🥳 Wohoo! Correct answer
Area formulas
Split integral at x = 0
Find absolute values in each region
Add to get area = 5/2
Region splitting errors
😢 Uh oh! Incorrect answer, Try again
Consider absolute values
Area formulas
Split integral at x = 0
Find absolute values in each region
Add to get area = 5/2
Region splitting errors
Degree and order of d²y/dx² = ∛(1+(dy/dx)²) are
🥳 Wohoo! Correct answer
Differential equation basics
Find highest derivative order
Find degree of highest derivative
Get order 2, degree 3
Order/degree confusion
😢 Uh oh! Incorrect answer, Try again
Check highest powers
Differential equation basics
Find highest derivative order
Find degree of highest derivative
Get order 2, degree 3
Order/degree confusion
Solution of x(dy/dx)-y=3 represents family of
🥳 Wohoo! Correct answer
Linear DE solutions
Rearrange to standard form
Use integrating factor
Get y = -3 + cx (straight lines)
Solution recognition errors
😢 Uh oh! Incorrect answer, Try again
Look for standard forms
Linear DE solutions
Rearrange to standard form
Use integrating factor
Get y = -3 + cx (straight lines)
Solution recognition errors
Integrating factor of (dy/dx)+y=(1+y)/x is
🥳 Wohoo! Correct answer
IF formula: e^∫P(x) dx
Find μ=e^∫P(x) dx where P(x)=1
Simplify expression
Get eˣ/x
Integration errors
😢 Uh oh! Incorrect answer, Try again
Standard formula for IF
IF formula: e^∫P(x) dx
Find μ=e^∫P(x) dx where P(x)=1
Simplify expression
Get eˣ/x
Integration errors
The value of C in Mean Value theorem for f(x)=x² in [2,4] is
🥳 Wohoo! Correct answer
MVT: f'(c)=[f(b)-f(a)]/[b-a]
Use MVT formula: f'(c)=[f(b)-f(a)]/[b-a]
Substitute f'(x)=2x and interval [2,4]
Solve to get c=3
Wrong interval substitution
😢 Uh oh! Incorrect answer, Try again
Draw graph to visualize
MVT: f'(c)=[f(b)-f(a)]/[b-a]
Use MVT formula: f'(c)=[f(b)-f(a)]/[b-a]
Substitute f'(x)=2x and interval [2,4]
Solve to get c=3
Wrong interval substitution
Point on curve y²=x where tangent makes angle π/4 with x-axis is
🥳 Wohoo! Correct answer
tanθ=dy/dx
Find dy/dx from y²=x
Use tan(π/4)=1 for slope
Solve to get (1/4,1/2)
Wrong differentiation
😢 Uh oh! Incorrect answer, Try again
Remember slope formula
tanθ=dy/dx
Find dy/dx from y²=x
Use tan(π/4)=1 for slope
Solve to get (1/4,1/2)
Wrong differentiation
The rate of change of sphere with respect to its surface area when radius is 4cm is
🥳 Wohoo! Correct answer
dV/dS=(dV/dr)÷(dS/dr)
Volume V=(4/3)πr³, Surface area S=4πr²
Find dV/dr=4πr² and dS/dr=8πr
Then dV/dS=(dV/dr)÷(dS/dr)=r/2=2
Not using chain rule correctly
😢 Uh oh! Incorrect answer, Try again
Rate of change involves derivatives
dV/dS=(dV/dr)÷(dS/dr)
Volume V=(4/3)πr³, Surface area S=4πr²
Find dV/dr=4πr² and dS/dr=8πr
Then dV/dS=(dV/dr)÷(dS/dr)=r/2=2
Not using chain rule correctly
∫[(x+3)eˣ/(x+4)²]dx equals
🥳 Wohoo! Correct answer
Integration by parts
Use substitution u=x+4
Apply partial fractions
Integrate to get eˣ/(x+4)+c
Wrong substitution
😢 Uh oh! Incorrect answer, Try again
Look for substitution
Integration by parts
Use substitution u=x+4
Apply partial fractions
Integrate to get eˣ/(x+4)+c
Wrong substitution
∫tanx/(cotx+tanx) dx equals
🥳 Wohoo! Correct answer
tanx·cotx=1
Write cotx as 1/tanx
Simplify fraction
Integrate from 0 to π/2
Wrong trigonometric ratios
😢 Uh oh! Incorrect answer, Try again
Convert to simpler terms
tanx·cotx=1
Write cotx as 1/tanx
Simplify fraction
Integrate from 0 to π/2
Wrong trigonometric ratios
Find the integral of 1/(e^(sin x) + 1) from -π/2 to π/2
🥳 Wohoo! Correct answer
Substitution method
Use substitution u=sinx
Notice symmetry in the interval [-π/2,π/2]
Evaluate to get π/2
Missing periodicity
😢 Uh oh! Incorrect answer, Try again
Look for symmetry in limits
Substitution method
Use substitution u=sinx
Notice symmetry in the interval [-π/2,π/2]
Evaluate to get π/2
Missing periodicity
The degree of differential equation [1+(dy/dx)²]=d²y/dx² is
🥳 Wohoo! Correct answer
Degree = power of highest order derivative
Highest power of derivative term is 2 in (dy/dx)²
Highest order derivative is d²y/dx²
Degree is 1 (no higher powers of highest order derivative)
Confusing order and degree
😢 Uh oh! Incorrect answer, Try again
Check highest power of highest order derivative
Degree = power of highest order derivative
Highest power of derivative term is 2 in (dy/dx)²
Highest order derivative is d²y/dx²
Degree is 1 (no higher powers of highest order derivative)
Confusing order and degree
∫₀.₂³·⁵ [x]dx equals
🥳 Wohoo! Correct answer
[x] gives greatest integer ≤x
Split integral at integer points
Use definition of greatest integer function
Add all parts
Wrong interval splitting
😢 Uh oh! Incorrect answer, Try again
Consider integer points
[x] gives greatest integer ≤x
Split integral at integer points
Use definition of greatest integer function
Add all parts
Wrong interval splitting
If y=e^(sin⁻¹(t²-1)) and x=e^(sec⁻¹(1/(t²-1))) then dy/dx is equal to
🥳 Wohoo! Correct answer
d/dx(sin⁻¹x) = 1/√(1-x²)
Take natural log of both equations
Differentiate implicitly
Simplify to get dy/dx = -y/x
Errors in chain rule application
😢 Uh oh! Incorrect answer, Try again
Use chain rule and implicit differentiation
d/dx(sin⁻¹x) = 1/√(1-x²)
Take natural log of both equations
Differentiate implicitly
Simplify to get dy/dx = -y/x
Errors in chain rule application
If xy = e^x-y then dy/dx is equal to
🥳 Wohoo! Correct answer
d/dx(ln x) = 1/x, d/dx(e^x) = e^x
Take natural log of both sides: ln(xy) = x-y
Differentiate both sides implicitly with respect to x
Simplify to get dy/dx = logx/(1+logx)²
Errors in implicit differentiation
😢 Uh oh! Incorrect answer, Try again
Use implicit differentiation and chain rule
d/dx(ln x) = 1/x, d/dx(e^x) = e^x
Take natural log of both sides: ln(xy) = x-y
Differentiate both sides implicitly with respect to x
Simplify to get dy/dx = logx/(1+logx)²
Errors in implicit differentiation
The value of ∫(e^x(1+x) dx)/(cos²(e^x.x)) is equal to
🥳 Wohoo! Correct answer
∫du/cos²u = tan(u)+c
Let u = e^x.x, then du = (1+x)e^x dx
Recognize integral form ∫du/cos²u
Apply standard integral formula to get tan(u)+c
Missing the substitution step
😢 Uh oh! Incorrect answer, Try again
Look for substitution possibility
∫du/cos²u = tan(u)+c
Let u = e^x.x, then du = (1+x)e^x dx
Recognize integral form ∫du/cos²u
Apply standard integral formula to get tan(u)+c
Missing the substitution step
The value of ∫(e^x(x²tan⁻¹x+tan⁻¹x+1))/(x²+1) dx is equal to
🥳 Wohoo! Correct answer
d/dx(e^x tan⁻¹x) = e^x tan⁻¹x + e^x/(1+x²)
Identify the form suitable for substitution
Use u = tan⁻¹x and appropriate substitution
Integrate to get e^x tan⁻¹x+c
Missing the pattern for substitution
😢 Uh oh! Incorrect answer, Try again
Look for term matching derivative pattern
d/dx(e^x tan⁻¹x) = e^x tan⁻¹x + e^x/(1+x²)
Identify the form suitable for substitution
Use u = tan⁻¹x and appropriate substitution
Integrate to get e^x tan⁻¹x+c
Missing the pattern for substitution
If x^m y^n =(x+y)^(m+n) then dy/dx is equal to
🥳 Wohoo! Correct answer
d/dx(ln(x^n)) = n/x
Take natural log of both sides
Differentiate implicitly with respect to x
Simplify to get dy/dx = y/x
Errors in implicit differentiation
😢 Uh oh! Incorrect answer, Try again
Use logarithmic differentiation
d/dx(ln(x^n)) = n/x
Take natural log of both sides
Differentiate implicitly with respect to x
Simplify to get dy/dx = y/x
Errors in implicit differentiation
The value of ∫(e^6logx - e^5logx) / (e^4logx - e^3logx) dx is equal to
🥳 Wohoo! Correct answer
e^(logx) = x
Simplify using laws of exponents: e^nlogx = x^n
Rewrite as ∫(x⁶ - x⁵)/(x⁴ - x³) dx
Integrate to get x³/3 + c
Not simplifying exponents properly
😢 Uh oh! Incorrect answer, Try again
Simplify exponential expressions first
e^(logx) = x
Simplify using laws of exponents: e^nlogx = x^n
Rewrite as ∫(x⁶ - x⁵)/(x⁴ - x³) dx
Integrate to get x³/3 + c
Not simplifying exponents properly
The differential coefficient of log₁₀x with respect to logₓ10 is
🥳 Wohoo! Correct answer
log_ab = 1/log_ba
Use chain rule: d/dx(log₁₀x) = 1/(x ln 10)
Express in terms of log_x10 = 1/log₁₀x
Simplify to get -(logx)²
Confusion with log properties
😢 Uh oh! Incorrect answer, Try again
Change of base formula helps
log_ab = 1/log_ba
Use chain rule: d/dx(log₁₀x) = 1/(x ln 10)
Express in terms of log_x10 = 1/log₁₀x
Simplify to get -(logx)²
Confusion with log properties
The slope of the tangent to the curve x=t² + 3t - 8, y=2t² - 2t - 5 at the point (2,-1) is
🥳 Wohoo! Correct answer
dy/dx = (dy/dt)/(dx/dt)
Find t when x=2: t² + 3t - 10 = 0
Solve to get t=2
Calculate dy/dx = (2t - 2)/(2t + 3) at t=2
Errors in parametric differentiation
😢 Uh oh! Incorrect answer, Try again
Use parametric differentiation
dy/dx = (dy/dt)/(dx/dt)
Find t when x=2: t² + 3t - 10 = 0
Solve to get t=2
Calculate dy/dx = (2t - 2)/(2t + 3) at t=2
Errors in parametric differentiation
The value of ∫(0 to π/2) sin¹⁰⁰⁰x/(sin¹⁰⁰⁰x + cos¹⁰⁰⁰x) dx is equal to
🥳 Wohoo! Correct answer
For symmetric integrals about π/4
Use substitution u = π/2 - x
Notice f(π/2 - x) = 1 - f(x)
Integrate to get π/4
Missing symmetry property
😢 Uh oh! Incorrect answer, Try again
Look for symmetry
For symmetric integrals about π/4
Use substitution u = π/2 - x
Notice f(π/2 - x) = 1 - f(x)
Integrate to get π/4
Missing symmetry property
If tan⁻¹(x²+y²) = α then dy/dx is equal to
🥳 Wohoo! Correct answer
d/dx(tan⁻¹u) = 1/(1+u²) × du/dx
Differentiate implicitly with respect to x
Use chain rule
Solve for dy/dx = -x/y
Errors in chain rule
😢 Uh oh! Incorrect answer, Try again
Use implicit differentiation
d/dx(tan⁻¹u) = 1/(1+u²) × du/dx
Differentiate implicitly with respect to x
Use chain rule
Solve for dy/dx = -x/y
Errors in chain rule
The solution for the differential equation (dy/y)+ (dx/y) = 0 is
🥳 Wohoo! Correct answer
Separable equation: dy/y = f(x) dx
Rearrange to get (dy/y) = -(dx/y)
Multiply both sides by y
Integrate to get xy = c
Not recognizing separable form
😢 Uh oh! Incorrect answer, Try again
Recognize separable form
Separable equation: dy/y = f(x) dx
Rearrange to get (dy/y) = -(dx/y)
Multiply both sides by y
Integrate to get xy = c
Not recognizing separable form
The order and degree of the differential equation 1+(dy/dx)² + sin(dy/dx)^(3/4) = (d²y/dx²) is
🥳 Wohoo! Correct answer
Degree is highest power of derivatives
Identify highest order derivative (2nd order)
Try to express in polynomial form
Note sin term prevents standard degree definition
Confusion between order and degree
😢 Uh oh! Incorrect answer, Try again
Order is highest derivative
Degree is highest power of derivatives
Identify highest order derivative (2nd order)
Try to express in polynomial form
Note sin term prevents standard degree definition
Confusion between order and degree
The rate of change of area of a circle with respect to its radius at r = 2 cms is
🥳 Wohoo! Correct answer
dA/dr = 2πr
Area A = πr²
Find dA/dr = 2πr
Substitute r = 2 to get 4π
Forgetting to multiply by π
😢 Uh oh! Incorrect answer, Try again
Use derivative of area formula
dA/dr = 2πr
Area A = πr²
Find dA/dr = 2πr
Substitute r = 2 to get 4π
Forgetting to multiply by π
Area lying between the curves y² = 2x and y = x is
🥳 Wohoo! Correct answer
Area = ∫(upper curve - lower curve) dx
Find points of intersection by solving y² = 2x and y = x
Set up definite integral: ∫(√2x - x) dx from 0 to 2
Evaluate to get 2/3 sq.units
Error in limits of integration
😢 Uh oh! Incorrect answer, Try again
Area between curves is integral of difference
Area = ∫(upper curve - lower curve) dx
Find points of intersection by solving y² = 2x and y = x
Set up definite integral: ∫(√2x - x) dx from 0 to 2
Evaluate to get 2/3 sq.units
Error in limits of integration
The value of ∫(√10-x)/(√x+√10-x) dx from 2 to 8 is
🥳 Wohoo! Correct answer
Integration by substitution
Substitute u = √x + √10-x
Simplify the integrand
Evaluate definite integral to get 3
Not identifying correct substitution
😢 Uh oh! Incorrect answer, Try again
Look for substitution possibility
Integration by substitution
Substitute u = √x + √10-x
Simplify the integrand
Evaluate definite integral to get 3
Not identifying correct substitution
∫(sinx/(3+4cos²x)) dx =
🥳 Wohoo! Correct answer
Integration of rational trig functions
Use substitution u=cosx
Rearrange to standard form
Integrate to get arctan form
Wrong substitution
😢 Uh oh! Incorrect answer, Try again
Look for rational function in cosx
Integration of rational trig functions
Use substitution u=cosx
Rearrange to standard form
Integrate to get arctan form
Wrong substitution
Integrating factor of xdy/dx - y = x⁴ - 3x is
🥳 Wohoo! Correct answer
IF = e^∫P(x) dx where P(x) is coefficient of y
Rewrite as dy/dx - y/x = x³ - 3
Identify as linear first order DE
IF = e^∫(-1/x) dx = 1/x
Not identifying correct form of DE
😢 Uh oh! Incorrect answer, Try again
Look for coefficient of y
IF = e^∫P(x) dx where P(x) is coefficient of y
Rewrite as dy/dx - y/x = x³ - 3
Identify as linear first order DE
IF = e^∫(-1/x) dx = 1/x
Not identifying correct form of DE
∫(-π to π) (1-x²)sinx·cos²x dx =
🥳 Wohoo! Correct answer
Odd function over symmetric interval
Use odd/even function properties
Observe integrand is odd
Integral over [-π,π] is 0
Not recognizing odd function
😢 Uh oh! Incorrect answer, Try again
Check symmetry of function
Odd function over symmetric interval
Use odd/even function properties
Observe integrand is odd
Integral over [-π,π] is 0
Not recognizing odd function
If y = f(x² + 2) and f'(3) = 5, then dy/dx at x = 1
🥳 Wohoo! Correct answer
• Chain rule: dy/dx = f'(u)·du/dx • u = x² + 2
Use chain rule
Substitute x = 1
Calculate derivative value
Wrong chain rule application
😢 Uh oh! Incorrect answer, Try again
Use function composition
• Chain rule: dy/dx = f'(u)·du/dx • u = x² + 2
Use chain rule
Substitute x = 1
Calculate derivative value
Wrong chain rule application
If x = acos³θ, y = asin³θ, then 1 + (dy/dx)²
🥳 Wohoo! Correct answer
• dy/dx = (dy/dθ)/(dx/dθ) • Parametric derivatives
Find dy/dx using parametric
Square and add 1
Simplify to sec² θ
Wrong parametric formulas
😢 Uh oh! Incorrect answer, Try again
Use parametric differentiation
• dy/dx = (dy/dθ)/(dx/dθ) • Parametric derivatives
Find dy/dx using parametric
Square and add 1
Simplify to sec² θ
Wrong parametric formulas
∫(1/(x²(x⁴+1)³/⁴)) dx equals
🥳 Wohoo! Correct answer
• Integration by substitution • Power rule
Let u = (1+x⁴)¹/⁴
Use substitution method
Integrate and simplify
Wrong substitution
😢 Uh oh! Incorrect answer, Try again
Consider substitution
• Integration by substitution • Power rule
Let u = (1+x⁴)¹/⁴
Use substitution method
Integrate and simplify
Wrong substitution
Evaluate ∫(0 to π/4) log((sinx+cosx)/cosx) dx
🥳 Wohoo! Correct answer
• Log properties • Integration formulas
Use log properties
Substitute u = tanx
Integrate and evaluate limits
Wrong integration limits
😢 Uh oh! Incorrect answer, Try again
Use integration by parts
• Log properties • Integration formulas
Use log properties
Substitute u = tanx
Integrate and evaluate limits
Wrong integration limits
If function g(x) defined by g(x) = x²⁰⁰/200 + x¹⁹⁹/199 + x¹⁹⁸/198 +...+ x²/2 + x + 5
🥳 Wohoo! Correct answer
• Power rule • Chain rule
Find g'(x) using differentiation
Put x = 0 in derivative
Verify g'(0) = 1
Missing terms in differentiation
😢 Uh oh! Incorrect answer, Try again
Consider term-by-term differentiation
• Power rule • Chain rule
Find g'(x) using differentiation
Put x = 0 in derivative
Verify g'(0) = 1
Missing terms in differentiation
If x=ct and y=c/t, find dy/dx at t=2
🥳 Wohoo! Correct answer
• dy/dx = (dy/dt)/(dx/dt) • Chain rule
Find dx/dt = c, dy/dt = -c/t²
Use chain rule dy/dx = (dy/dt)/(dx/dt)
At t=2: dy/dx = -1/4
Wrong chain rule
😢 Uh oh! Incorrect answer, Try again
Use parametric differentiation
• dy/dx = (dy/dt)/(dx/dt) • Chain rule
Find dx/dt = c, dy/dt = -c/t²
Use chain rule dy/dx = (dy/dt)/(dx/dt)
At t=2: dy/dx = -1/4
Wrong chain rule
Balloon remains spherical, 10 cc gas/sec pumped, rate of radius increase at r=15 cm
🥳 Wohoo! Correct answer
• V = 4/3πr³ • Chain rule
Use V = 4/3πr³
dV/dt = 10, dV/dr = 4πr²
dr/dt = 1/90π at r=15
Wrong chain rule
😢 Uh oh! Incorrect answer, Try again
Consider volume-radius relation
• V = 4/3πr³ • Chain rule
Use V = 4/3πr³
dV/dt = 10, dV/dr = 4πr²
dr/dt = 1/90π at r=15
Wrong chain rule
∫(sin²x)/(1+cosx) dx
🥳 Wohoo! Correct answer
• sin²x = 1-cos²x • Integration by substitution
Use substitution u = 1+cosx
Simplify integrand
Get x - sinx + C
Wrong substitution
😢 Uh oh! Incorrect answer, Try again
Consider trig identities
• sin²x = 1-cos²x • Integration by substitution
Use substitution u = 1+cosx
Simplify integrand
Get x - sinx + C
Wrong substitution
∫eˣ[(1+sinx)/(1+cosx)] dx
🥳 Wohoo! Correct answer
• Integration by parts • Trig substitution
Use substitution u = tan(x/2)
Simplify expression
Integrate to get result
Wrong integration technique
😢 Uh oh! Incorrect answer, Try again
Consider trig substitution
• Integration by parts • Trig substitution
Use substitution u = tan(x/2)
Simplify expression
Integrate to get result
Wrong integration technique
If (x+1)²/(x³+x) = A/x + (Bx+C)/(x²+1), then cosec⁻¹(1/A) + cot⁻¹(1/B) + sec⁻¹C =
🥳 Wohoo! Correct answer
• Partial fraction decomposition • Inverse trig sum formulas
Equate coefficients
Find A=1, B=0, C=2
Sum equals 0
Wrong decomposition
😢 Uh oh! Incorrect answer, Try again
Use partial fractions
• Partial fraction decomposition • Inverse trig sum formulas
Equate coefficients
Find A=1, B=0, C=2
Sum equals 0
Wrong decomposition
Two curves x³-3xy²+2=0 and 3x²y-y³=2
🥳 Wohoo! Correct answer
• Implicit differentiation • Perpendicular slopes
Find dy/dx for both curves
Find intersection points
Show perpendicular slopes
Wrong differentiation
😢 Uh oh! Incorrect answer, Try again
Consider curve derivatives
• Implicit differentiation • Perpendicular slopes
Find dy/dx for both curves
Find intersection points
Show perpendicular slopes
Wrong differentiation
If x is real, minimum value of x² - 8x + 17 is
🥳 Wohoo! Correct answer
• f'(x) = 0 for minimum • Complete square formula
Complete square: (x-4)² + 1
Find derivative
Minimum at x = 4 gives 1
Wrong differentiation
😢 Uh oh! Incorrect answer, Try again
Consider quadratic form
• f'(x) = 0 for minimum • Complete square formula
Complete square: (x-4)² + 1
Find derivative
Minimum at x = 4 gives 1
Wrong differentiation
∫(π/4 to -π/2) dx/(1+cos2x) equals
🥳 Wohoo! Correct answer
• ∫dx/(1+cosx) formula • Substitution method
Use substitution u = 2x
Apply integral formula
Evaluate at limits
Wrong substitution
😢 Uh oh! Incorrect answer, Try again
Consider trigonometric forms
• ∫dx/(1+cosx) formula • Substitution method
Use substitution u = 2x
Apply integral formula
Evaluate at limits
Wrong substitution
Order of differential equation of all circles of radius 'a' is
🥳 Wohoo! Correct answer
• (x-h)² + (y-k)² = a² • Order definition
Write general circle equation
Find differential equation
Verify order is 2
Wrong equation formation
😢 Uh oh! Incorrect answer, Try again
Consider elimination method
• (x-h)² + (y-k)² = a² • Order definition
Write general circle equation
Find differential equation
Verify order is 2
Wrong equation formation
Solution of x(dy/dx) + 2y = x²
🥳 Wohoo! Correct answer
• IF = e^∫P(x) dx • Linear DE formula
Rearrange to standard form
Multiply by integrating factor
Solve for y
Wrong integrating factor
😢 Uh oh! Incorrect answer, Try again
Consider linear DE solution
• IF = e^∫P(x) dx • Linear DE formula
Rearrange to standard form
Multiply by integrating factor
Solve for y
Wrong integrating factor
Solution of e^(dy/dx) = x+1, y(0)=3 is
🥳 Wohoo! Correct answer
Integration of e^x
Separate variables
Integrate both sides
Use initial condition
Wrong integration
😢 Uh oh! Incorrect answer, Try again
Use separation of variables
Integration of e^x
Separate variables
Integrate both sides
Use initial condition
Wrong integration
Family of curves whose x,y intercepts of tangent at any point are double x,y coordinates of that point is
🥳 Wohoo! Correct answer
Point-slope form of tangent
Use point-slope form
Apply intercept condition
Get xy=C family
Wrong differential equation
😢 Uh oh! Incorrect answer, Try again
Think about ratio property
Point-slope form of tangent
Use point-slope form
Apply intercept condition
Get xy=C family
Wrong differential equation
If y = asinx + bcosx, then y² + (dy/dx)² is a
🥳 Wohoo! Correct answer
Differentiation rules
Find dy/dx = acosxbsinx
Calculate y² + (dy/dx)²
Show result is a² + b² (constant)
Not recognizing constant pattern
😢 Uh oh! Incorrect answer, Try again
Complete the square method
Differentiation rules
Find dy/dx = acosxbsinx
Calculate y² + (dy/dx)²
Show result is a² + b² (constant)
Not recognizing constant pattern
If f(x) = 1 + nx + n(n-1)x²/2 + n(n-1)(n-2)x³/6 + ... + xⁿ then f ''(1)
🥳 Wohoo! Correct answer
Differentiation formulas
Find f'(x) by term by term differentiation
Find f''(x)
Evaluate at x = 1
Missing pattern recognition
😢 Uh oh! Incorrect answer, Try again
Use binomial expansion pattern
Differentiation formulas
Find f'(x) by term by term differentiation
Find f''(x)
Evaluate at x = 1
Missing pattern recognition
If u = sin⁻¹(2x/(1+x²)) and v = tan⁻¹(2x/(1-x²)) then du/dv is
🥳 Wohoo! Correct answer
Inverse trig derivatives
Find du/dx using chain rule
Find dv/dx using chain rule
Divide du/dx by dv/dx
Chain rule application errors
😢 Uh oh! Incorrect answer, Try again
Use chain rule and inverse trig derivatives
Inverse trig derivatives
Find du/dx using chain rule
Find dv/dx using chain rule
Divide du/dx by dv/dx
Chain rule application errors
A particle moves along the curve x²/16 + y²/4 = 1. When the rate of change of abscissa is 4 times that of its ordinate, then the quadrant in which the particle lies is
🥳 Wohoo! Correct answer
Parametric differentiation
Find dx/dt and dy/dt relationship
Use given condition dx/dt = 4dy/dt
Determine quadrant from signs
Sign confusion in quadrants
😢 Uh oh! Incorrect answer, Try again
Use implicit differentiation
Parametric differentiation
Find dx/dt and dy/dt relationship
Use given condition dx/dt = 4dy/dt
Determine quadrant from signs
Sign confusion in quadrants
∫√(cosecx - sinx) dx =
🥳 Wohoo! Correct answer
Integration formulas
Rationalize using cosecx = 1/sinx
Simplify under square root
Integrate to get 2√sinx + C
Missing rationalization
😢 Uh oh! Incorrect answer, Try again
Rationalize and simplify first
Integration formulas
Rationalize using cosecx = 1/sinx
Simplify under square root
Integrate to get 2√sinx + C
Missing rationalization
∫[(sin(5x/2)) / (sin(x/2))] dx =
🥳 Wohoo! Correct answer
Integration of trigonometric functions
Simplify fraction first
Break into partial fractions
Integrate term by term
Not simplifying before integration
😢 Uh oh! Incorrect answer, Try again
Look for cancellation in fraction
Integration of trigonometric functions
Simplify fraction first
Break into partial fractions
Integrate term by term
Not simplifying before integration
A circular plate of radius 5 cm is heated. Due to expansion, its radius increases at the rate of 0.05 cm/sec. The rate at which its area is increasing when the radius is 5.2 cm is
🥳 Wohoo! Correct answer
Area change formula
Use A = πr²
dA/dt = 2πr·dr/dt
Substitute values to get 0.52cm²/sec
Unit conversion errors
😢 Uh oh! Incorrect answer, Try again
Use chain rule for differentiation
Area change formula
Use A = πr²
dA/dt = 2πr·dr/dt
Substitute values to get 0.52cm²/sec
Unit conversion errors
The distance 's' in meters travelled by a particle in 't' seconds is given by s = 2t³/3 - 18t + 5/3. The acceleration when the particle comes to rest is
🥳 Wohoo! Correct answer
Kinematics formulas
Find v = ds/dt = 2t² - 18
Find a = dv/dt = 4t
At rest v = 0, solve for t = 3
Wrong time substitution
😢 Uh oh! Incorrect answer, Try again
Differentiate twice for acceleration
Kinematics formulas
Find v = ds/dt = 2t² - 18
Find a = dv/dt = 4t
At rest v = 0, solve for t = 3
Wrong time substitution
Integrate 0 to π ∫(xtanx)/(secx·cosecx) dx =
🥳 Wohoo! Correct answer
Integration by parts formula; trig identities
Simplify integrand: xtanx/(secx·cosecx) = xsin²x
Integrate by parts: u = x, dv = sin²x dx
Evaluate from 0 to π/2 to get 2π/4
Not recognizing simplification of secx·cosecx
😢 Uh oh! Incorrect answer, Try again
Use trigonometric simplification first
Integration by parts formula; trig identities
Simplify integrand: xtanx/(secx·cosecx) = xsin²x
Integrate by parts: u = x, dv = sin²x dx
Evaluate from 0 to π/2 to get 2π/4
Not recognizing simplification of secx·cosecx
∫1/(1+3sin²x+8cos²x) dx =
🥳 Wohoo! Correct answer
Integration of rational functions of sin and cos
Convert to terms of tan²x using sin²x = tan²x/(1+tan²x)
Simplify to form 1/(a+btan²x)
Integrate to get (1/6)tan⁻¹(2tanx/3) + C
Wrong trigonometric substitution
😢 Uh oh! Incorrect answer, Try again
Use tan²x substitution
Integration of rational functions of sin and cos
Convert to terms of tan²x using sin²x = tan²x/(1+tan²x)
Simplify to form 1/(a+btan²x)
Integrate to get (1/6)tan⁻¹(2tanx/3) + C
Wrong trigonometric substitution
Integrate -2 to 0 ∫[x³ + 3x² + 3x + 3 + (x+1)cos(x+1)]dx =
🥳 Wohoo! Correct answer
Integration formulas; Integration by parts
Split integral into polynomial and trigonometric parts
Integrate polynomial term by term and trigonometric part by parts
Evaluate at limits -2 to 0
Integration by parts error
😢 Uh oh! Incorrect answer, Try again
Split into simpler integrals
Integration formulas; Integration by parts
Split integral into polynomial and trigonometric parts
Integrate polynomial term by term and trigonometric part by parts
Evaluate at limits -2 to 0
Integration by parts error
Find the degree of the differential equation:1 + (dy/dx)² + (d²y/dx²)² = ∛(d²y/dx² + 1)
🥳 Wohoo! Correct answer
Order and degree of differential equations
Identify highest order derivative
Count power of highest derivative
Degree is 6 due to cube of term with second derivative
Confusing order with degree
😢 Uh oh! Incorrect answer, Try again
Look for highest power of highest derivative
Order and degree of differential equations
Identify highest order derivative
Count power of highest derivative
Degree is 6 due to cube of term with second derivative
Confusing order with degree
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