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Wize University Calculus 1 Textbook > Integration Techniques

Exam-Like Practice Questions

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

Find∫04xf′(x)dx  if f(4)=−3\int_0^4xf'\left(x\right)dx\ \ if\ f\left(4\right)=-3 and∫04f(x)dx=−3\int_0^4f\left(x\right)dx=-3
Evaluate ∫1e(ln⁡x)2dx\int_1^e\left(\ln x\right)^2dx
Evaluate∫arctan⁡(1x)dx\int_{ }^{ }\arctan(\frac{1}{x})dx
Evaluate ∫x2e−xdx\displaystyle\int x^2e^{-x}dx


Practice Question

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

Practice Question

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

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Practice Question

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

Practice: Trig Sub

Evaluate ∫01x4−x2dx\displaystyle\int_0^1\frac{x}{\sqrt{4-x^2}}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}.

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

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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. }


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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. }



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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

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Use The Trapezoid Rule to approximate

∫251x−1dx\int_2^5\frac{1}{x-1}dx

using 4 subintervals
Approximate ∫151+x2 dx\displaystyle\int\limits_{1}^{5}\sqrt{1+x^{2}}\,dx using the Trapezoidal Rule with n=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