1.7 Chapter 1 Review

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Sample Tests for Chapter 1

Sample Test 1a

(problem 1) Compute $\displaystyle \int \frac {\ln ^3(x)}{x} \, dx$

u-sub with $u = \ln (x)$

(problem 2) Compute $\displaystyle \int x^2(x^3 + 2)^7 \, dx$

u-sub with $u = x^3 + 2$

(problem 3) Compute $\displaystyle \int x^2\cos (4x) \, dx$

IBP $\int u dv = uv - \int v du$

LIATE

Need IBP twice- try Tabular Integration

(problem 4) Compute $\displaystyle \int \tan ^{-1}(x) \, dx$

IBP $\int u dv = uv - \int v du$

LIATE

$\displaystyle \frac {d}{dx} \tan ^{-1} (x) = \frac {1}{1+x^2}$

(problem 5) Compute $\displaystyle \int \sin ^4(x) \, dx$

Use the half angle formula twice

$\sin ^2 \theta = \frac {1-\cos (2\theta )}{2} \quad \mbox {and} \cos ^2 \theta = \frac {1+\cos (2\theta )}{2}$

(problem 6) Compute $\displaystyle \int \tan ^4(x) \sec ^6(x) \, dx$

set aside $\sec ^2(x)$ to prepare for u-sub

convert remaining secants into tangents using $1+ \tan ^2(x) = \sec ^2(x)$

(problem 7) Compute $\displaystyle \int x^3 \sqrt {1-x^2} \, dx$

trig sub with $x= \cos \theta$

(problem 8) Compute $\displaystyle \int \frac {1}{ \sqrt {x^2 - 4}} \, dx$

trig sub with $x= 2\sec \theta$

$\displaystyle \int \sec \theta , d \theta = \ln |\sec \theta + \tan \theta |$

(problem 9) Compute $\displaystyle \int \frac {3}{x^2 + x} \, dx$

Partial Fraction (distinct linear factors)

(problem 10) Compute $\displaystyle \int \frac {3x-4}{x^3 +8x^2 +16x} \, dx$

Partial Fraction (repeated linear factor)

$x^2 + 8x + 16 = (x+4)^2$

(problem 11) Compute $\displaystyle \int _1^\infty 3e^{-2x} \, dx$

replace $\infty$ with $t$ and take the limit as $t$ approaches $\infty$

$\displaystyle \int e^{ax} dx = \frac {1}{a} e^{ax} + C$

to calculate the limit, remove the negative exponent

(problem 12) Compute $\displaystyle \int _{0}^2 \frac {5}{(x-2)^2} \, dx$

replace $2$ with $t$ and take the limit as $t$ approaches $2$ from the left

to calculate the limit, remove the negative exponent

Sample Test 1b

(problem 1) Compute $\displaystyle \int \frac {x}{(x^2+1)^3} \, dx$

answer: $\displaystyle -\frac {1}{4(x^2+1)^2} + C$

(problem 2) Compute $\displaystyle \int \cot (x) \, dx$

Write $\cot (x)$ as $\frac {\cos (x)}{\sin (x)}$ and do u-sub with $u = \sin (x)$

answer: $\ln |\sin (x)| + C$

(problem 3) Compute $\displaystyle \int x^2\ln (x) \, dx$

answer: $\displaystyle \frac {x^3}{3} \ln (x) - \frac {x^3}{9} + C$

(problem 4) Compute $\displaystyle \int e^x \cos (x) \, dx$

use integration by parts twice to get back to the original integral ‘I’

answer: $I = \frac 12 e^x \cos x + \frac 12 e^x \sin x + C$

(problem 5) Compute $\displaystyle \int \sin ^3(2x) \cos ^3(2x) \, dx$

write $\cos ^3(2x)$ as $\cos ^2(2x) \cdot \cos (2x)$

let $u = \sin (2x)$ , then $du = 2\cos (2x) \, dx$

answer: $\displaystyle \frac 18 \sin ^4 (2x) - \frac {1}{12} \sin ^6 (2x) + C$

(problem 6) Compute $\displaystyle \int \sec ^5(x) \tan (x) \, dx$

answer: $\displaystyle \frac {\sec ^5(x)}{5} + C$

(problem 7) Compute $\displaystyle \int \frac {x^3}{ \sqrt {1+x^2}} \, dx$

use trig sub with $x = \tan \theta$ and $dx = \sec ^2 \theta \, d\theta$

(problem 8) Compute $\displaystyle \int \frac {1}{x^2 \sqrt {x^2 - 25}} \, dx$

use trig sub with $x = 5\sec \theta$ and $dx = 5 \sec \theta \tan \theta \, d\theta$

(problem 9) Compute $\displaystyle \int \frac {x^2 - 2x+3}{x^3-x} \, dx$

$x^2 - 1$ is a difference of two squares so it factors as: $(x-1)(x+1)$

(problem 10) Compute $\displaystyle \int \frac {x^2 + 5x + 2}{x^3 + 16x} \, dx$

$x^2 + 16$ is an irreducible quadratic

$\int \frac {1}{x^2 + a^2} \, dx = \frac {1}{a} \tan ^{-1}\left (\frac {x}{a}\right )$

(problem 11) Compute $\displaystyle \int _4^\infty \frac {4}{x^4} \, dx$

the integral converges to $1/48$

(problem 12) Compute $\displaystyle \int _1^5 \frac {3}{\sqrt {x-1}} \, dx$

the integrand has a vertical asymptote at $x = 1$

the integral converges to 12