Exam-Like Practice Questions

Evaluate ∫xsec⁡xtan⁡x  ⁣dx{\displaystyle\int}x\sec x\tan x\de{x}∫xsecxtanx dx.

x sec⁡x−tanx+Cx\ \sec x-tanx+Cx secx−tanx+C

secx−ln⁡∣sec⁡x+tan⁡x∣+Csec x-\ln\left|\sec x+\tan x\right|+Csecx−ln∣secx+tanx∣+C

x sec⁡x−secxtanx+Cx\ \sec x-sec xtan x+Cx secx−secxtanx+C

x sec⁡x−ln⁡∣sec⁡x+tan⁡x∣+Cx\ \sec x-\ln\left|\sec x+\tan x\right|+Cx secx−ln∣secx+tanx∣+C

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Find∫04xf′(x)dx  if f(4)=−3\int_0^4xf'\left(x\right)dx\ \ if\ f\left(4\right)=-3∫04​xf′(x)dx  if f(4)=−3 and∫04f(x)dx=−3\int_0^4f\left(x\right)dx=-3∫04​f(x)dx=−3

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Evaluate ∫1e(ln⁡x)2dx\int_1^e\left(\ln x\right)^2dx∫1e​(lnx)2dx 

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Evaluate∫arctan⁡(1x)dx\int_{ }^{ }\arctan(\frac{1}{x})dx∫​arctan(x1​)dx

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Evaluate ∫x2e−xdx\displaystyle\int x^2e^{-x}dx∫x2e−xdx

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Practice Question 
Compute ∫0π/4sec⁡6x  ⁣dx{\displaystyle\int_0^{\pi/4}}\sec^6x\de{x}∫0π/4​sec6x dx.

Practice Question
 Find∫sin⁡(15x)cos⁡(4x)  ⁣dx{\displaystyle\int}\sin(15x)\cos(4x)\de{x}∫sin(15x)cos(4x) dx.

Use upper case

C

for any constants

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Practice Question
Evaluate∫sec⁡3x   ⁣dx{\displaystyle\int}\sec^3x \space\de{x}∫sec3x  dx.
 

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Practice: Trig Sub
Evaluate ∫01x4−x2dx\displaystyle\int_0^1\frac{x}{\sqrt{4-x^2}}dx∫01​4−x2​x​dx

*You can technically use a u-substitution for this question, but try to use trig sub for additional practice

Practice Question
Find ∫1x2−2x−3  ⁣dx{\displaystyle\int}\frac{1}{\sqrt{x^2-2x-3}}\de{x}∫x2−2x−3​1​ dx.

ln⁡∣(x−1)2−42∣+C\ln\left|\frac{\sqrt{\left(x-1\right)^2-4}}{2}\right|+Cln​2(x−1)2−4​​​+C

ln⁡∣x−12+(x−1)2−42∣+C\ln\left|\frac{x-1}{2}+\frac{\sqrt{\left(x-1\right)^2-4}}{2}\right|+Cln​2x−1​+2(x−1)2−4​​​+C

ln⁡∣x−12∣+C\ln\left|\frac{x-1}{2}\right|+Cln​2x−1​​+C

ln⁡∣x−12+(x−1)2−42∣+C\ln\left|\frac{\sqrt{x-1}}{2}+\frac{\left(x-1\right)^2-4}{2}\right|+Cln​2x−1​​+2(x−1)2−4​​+C

ln⁡∣(x−1)2−42∣+C\ln\left|\frac{\left(x-1\right)^2-4}{2}\right|+Cln​2(x−1)2−4​​+C

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∫xx2−2x+3 dx.\bcb{
\displaystyle \int \frac{x}{x^2-2x+3}\ {\rm d}x.
}∫x2−2x+3x​ dx. 

Answer

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Set up the partial fraction decomposition; solve for the constants, but do not integrate: 
∫1−x+2x2−x3x(x2+1)2 dx.\bcb{
\displaystyle \int \frac{1-x+2x^2-x^3}{x(x^2+1)^2}\ {\rm d}x.
}∫x(x2+1)21−x+2x2−x3​ dx.

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Set up the partial fraction decomposition; solve for the constants, but do not integrate: 
∫4xx3−x2−x+1 dx.\bcb{
\displaystyle \int \frac{4x}{x^3-x^2-x+1}\ {\rm d}x.
}∫x3−x2−x+14x​ dx.

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Find∫x2−2x−3(x2−9)(x−4)dx\displaystyle\int\frac{x^2-2x-3}{\left(x^2-9\right)\left(x-4\right)}dx∫(x2−9)(x−4)x2−2x−3​dx

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Use The Trapezoid Rule to approximate 

∫251x−1dx\int_2^5\frac{1}{x-1}dx∫25​x−11​dx

using 4 subintervals 

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Approximate ∫151+x2 dx\displaystyle\int\limits_{1}^{5}\sqrt{1+x^{2}}\,dx1∫5​1+x2​dx  using the Trapezoidal Rule with n=8n=8n=8 (answer correct to 2 decimal places).
 

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Approximate the following using a Simpson's Rule approximation with 6 subintervals. 

∫0πsin⁡xdx\int_0^{\pi}\sin xdx∫0π​sinxdx

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Wize AP Calculus (BC) Textbook > Integration Techniques

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Practice: Trig Sub

Practice Question

Set up the partial fraction decomposition; solve for the constants, but do not integrate:

$\bcb{ \displaystyle \int \frac{1-x+2x^2-x^3}{x(x^2+1)^2}\ {\rm d}x. }$

Set up the partial fraction decomposition; solve for the constants, but do not integrate:

$\bcb{ \displaystyle \int \frac{4x}{x^3-x^2-x+1}\ {\rm d}x. }$

Wize AP Calculus (BC) Textbook > Integration Techniques

Exam-Like Practice Questions

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Practice Question

Practice Question

Practice Question

Practice: Trig Sub

Practice Question

Set up the partial fraction decomposition; solve for the constants, but do not integrate:

∫1−x+2x2−x3x(x2+1)2 dx.\bcb{ \displaystyle \int \frac{1-x+2x^2-x^3}{x(x^2+1)^2}\ {\rm d}x. }∫x(x2+1)21−x+2x2−x3​ dx.

Set up the partial fraction decomposition; solve for the constants, but do not integrate:

∫4xx3−x2−x+1 dx.\bcb{ \displaystyle \int \frac{4x}{x^3-x^2-x+1}\ {\rm d}x. }∫x3−x2−x+14x​ dx.

$\bcb{ \displaystyle \int \frac{1-x+2x^2-x^3}{x(x^2+1)^2}\ {\rm d}x. }$

$\bcb{ \displaystyle \int \frac{4x}{x^3-x^2-x+1}\ {\rm d}x. }$