Basic Technical Mathematics with Calculus

Allyn J. Washington, Richard S. Evans

Chapter 28

Methods of Integration - all with Video Answers

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Section 1

The General Power Formula

Problem 1

Make the given changes in the indicated examples of this section and then solve the given problems.
In Example $1,$ change $\sin ^{3} x \cos x$ to $\cos ^{3} x \sin x$ and then integrate.

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Problem 2

Make the given changes in the indicated examples of this section and then solve the given problems.
In Example $3,$ change $\ln x$ to $\ln x^{2}$ and then integrate.

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Problem 3

Integrate each of the functions.
$$\int\left(x^{2}+1\right)^{3}(2 x d x)$$

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Problem 4

Integrate each of the functions.
$$\int\left(x^{3}-2\right)^{6}\left(3 x^{2} d x\right)$$

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Problem 5

Integrate each of the functions.
$$\int \sin ^{4} x \cos x d x$$

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Problem 6

Integrate each of the functions.
$$\int \cos ^{5} x(-\sin x d x)$$

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Problem 7

Integrate each of the functions.
$$\int 0.4 \sqrt{\cos \theta} \sin \theta d \theta$$

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Problem 8

Integrate each of the functions.
$$\int 8 \sin ^{1 / 3} x \cos x d x$$

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Problem 9

Integrate each of the functions.
$$\int 4 \tan ^{2} x \sec ^{2} x d x$$

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Problem 10

Integrate each of the functions.
$$\int 3 \sec ^{3} x(\sec x \tan x) d x$$

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Problem 11

Integrate each of the functions.
$$\int_{0}^{\pi / 8} \frac{\cos 2 x}{\csc 2 x} d x$$

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Problem 12

Integrate each of the functions.
$$\int_{\pi / 6}^{\pi / 4} 3 \sqrt{\cot x} \csc ^{2} x d x$$

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Problem 13

Integrate each of the functions.
$$\int\left(\sin ^{-1} x\right)^{3}\left(\frac{d x}{\sqrt{1-x^{2}}}\right)$$

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Problem 14

Integrate each of the functions.
$$\int \frac{20\left(\cos ^{-1} 2 t\right)^{4} d t}{\sqrt{1-4 t^{2}}}$$

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Problem 15

Integrate each of the functions.
$$\int \frac{5 \tan ^{-1} 5 x}{25 x^{2}+1} d x$$

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Problem 16

Integrate each of the functions.
$$\int \frac{\sin ^{-1} 4 x d x}{\sqrt{1-16 x^{2}}}$$

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Problem 17

Integrate each of the functions.
$$\int[\ln (x+1)]^{2} \frac{d x}{x+1}$$

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Problem 18

Integrate each of the functions.
$$\int 0.8(3+2 \ln u)^{3} \frac{d u}{u}$$

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Problem 19

Integrate each of the functions.
$$\int_{0}^{1 / 2} \frac{\ln (2 x+3)}{4 x+6} d x$$

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Problem 20

Integrate each of the functions.
$$\int_{1}^{e} \frac{(1-2 \ln x) d x}{4 x}$$

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Problem 21

Integrate each of the functions.
$$\int 3\left(4+e^{x}\right)^{3} e^{x} d x$$

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Problem 22

Integrate each of the functions.
$$\int 2 \sqrt{1-e^{-x}}\left(-e^{-x} d x\right)$$

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Problem 23

Integrate each of the functions.
$$\int \frac{4 e^{2 t} d t}{\left(1-e^{2 t}\right)^{3}}$$

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Problem 24

Integrate each of the functions.
$$\int \frac{\left(1+3 e^{-2 x}\right)^{4} d x}{6 e^{2 x}}$$

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Problem 25

Integrate each of the functions.
$$\int\left(1+\sec ^{2} x\right)^{4}\left(\sec ^{2} x \tan x d x\right)$$

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Problem 26

Integrate each of the functions.
$$\int\left(e^{x}+e^{-x}\right)^{1 / 4}\left(e^{x}-e^{-x}\right) d x$$

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Problem 27

Integrate each of the functions.
$$\int_{\pi / 6}^{\pi / 4}(1+\cot x)^{2} \csc ^{2} x d x$$

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Problem 28

Integrate each of the functions.
$$\int_{\pi / 3}^{\pi / 2} \frac{2 \sin \theta d \theta}{\sqrt{1+\cos \theta}}$$

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Problem 29

Rewrite the given integrals so that they fit the form $\int u^{n} d u,$ and identify $u, n,$ and $d u$.
$$\int \sec ^{5} x \sin x d x$$

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Problem 30

Rewrite the given integrals so that they fit the form $\int u^{n} d u,$ and identify $u, n,$ and $d u$.
$$\int \frac{\tan ^{3} x d x}{\cos ^{2} x}$$

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Problem 31

Rewrite the given integrals so that they fit the form $\int u^{n} d u,$ and identify $u, n,$ and $d u$.
$$\int \frac{d x}{x \ln ^{2} x}$$

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Problem 32

Rewrite the given integrals so that they fit the form $\int u^{n} d u,$ and identify $u, n,$ and $d u$.
$$\int \frac{e^{-1 / x}}{x^{2}} d x$$

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Problem 33

Solve the given problems by integration.
Find the first-quadrant area under the curve of $y=\ln ^{2} x / x$ from $x=1$ to $x=4$.

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Problem 34

Solve the given problems by integration.
Find the area under $y=6 \sin ^{2} x \cos x$ from $x=0$ to $x=\pi / 2$.

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Problem 35

Solve the given problems by integration.
Find the volume generated when the first-quadrant area bounded by $y=e^{x}$ and $x=2$ is rotated about the $x$ -axis. [Hint: After setting up the integral, rewrite $\left.\left(e^{x}\right)^{2} \text { as }\left(e^{x}\right)\left(e^{x}\right) .\right]$

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Problem 36

Solve the given problems by integration.
Find the volume generated if the first-quadrant region bounded by $y=e^{2 x}$ and $x=1$ is revolved about the $x$ -axis. See the hint in Exercise 35.

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Problem 37

Solve the given problems by integration.
Find the area under the curve $y=\frac{1+\tan ^{-1} 2 x}{1+4 x^{2}}$ from $x=0$ to $x=2$.

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Problem 38

Solve the given problems by integration.
Find the first-quadrant area bounded by $y=\frac{\ln (4 x+1)}{4 x+1}$ and $x=5$.

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Problem 39

Solve the given problems by integration.
The general expression for the slope of a given curve is $(\ln x)^{2} / x$ If the curve passes through $(1,2),$ find its equation.

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Problem 40

Solve the given problems by integration.
Find an equation of the curve for which $d y / d x=(1+\tan 2 x)^{2} \sec ^{2} 2 x$ if the curve passes through (2,1).

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Problem 41

Solve the given problems by integration.
In the development of the expression for the total pressure $P$ on a wall due to molecules with mass $m$ and velocity $v$ striking the wall, the equation $P=m n v^{2} \int_{0}^{\pi / 2} \sin \theta \cos ^{2} \theta d \theta$ is found. The symbol $n$ represents the number of molecules per unit volume, and $\theta$ represents the angle between a perpendicular to the wall and the direction of the molecule. Find the expression for $P$

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Problem 42

Solve the given problems by integration.
The solar energy $E$ passing through a hemispherical surface per unit time, per unit area, is $E=2 \pi I \int_{0}^{\pi / 2} \cos \theta \sin \theta d \theta,$ where $I$ is the solar intensity and $\theta$ is the angle at which it is directed (from the perpendicular). Evaluate this integral.

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Problem 43

Solve the given problems by integration.
After an electric power interruption, the current $i$ in a circuit is given by $i=3\left(1-e^{-t}\right)^{2}\left(e^{-t}\right),$ where $t$ is the time. Find the expression for the total electric charge $q$ to pass a point in the circuit if $q=0$ for $t=0$.

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Problem 44

Solve the given problems by integration.
A space vehicle is launched vertically from the ground such that its velocity $v(\text { in } \mathrm{km} / \mathrm{s})$ is given by $v=\left[\ln ^{2}\left(t^{3}+1\right)\right] \frac{t^{2}}{t^{3}+1},$ where $t$ is the time (in s). Find the altitude of the vehicle after $10.0 \mathrm{s}$.

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