Basic Technical Mathematics with Calculus

Allyn J. Washington, Richard S. Evans

Chapter 28 Methods of Integration - all with Video Answers

Educators

Section 5

Other Trigonometric Forms

01:44

Problem 1

In Exercises 1 and 2 , answer the given questions related to the indicated examples of this section.In Example $3,$ change $2 x$ to $3 x$ and then integrate.

Rukhmani J.

Numerade Educator

Problem 2

Answer the given questions related to the indicated examples of this section.
In Example $5,$ change the exponent of 5 to a 3 and then integrate.

Check back soon!

Problem 3

Integrate each of the given functions.
$$\int \sin ^{2} x \cos x d x$$

Check back soon!

Problem 4

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

Check back soon!

Problem 5

Integrate each of the given functions.
$$\int \sin ^{3} 2 x d x$$

Check back soon!

Problem 6

Integrate each of the given functions.
$$\int 3 \cos ^{3} T d T$$

Check back soon!

Problem 7

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

Check back soon!

Problem 8

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

Check back soon!

Problem 9

Integrate each of the given functions.
$$\int_{0}^{\pi / 2} \sin ^{4} x \cos ^{3} x d x$$

Check back soon!

Problem 10

Integrate each of the given functions.
$$\int_{\pi / 3}^{\pi / 2} 10 \sin t(1-\cos 2 t)^{2} d t$$

Check back soon!

Problem 11

Integrate each of the given functions.
$$\int \sin ^{2} x d x$$

Check back soon!

Problem 12

Integrate each of the given functions.
$$\int 2 \cos ^{2} 2 x d x$$

Check back soon!

Problem 13

Integrate each of the given functions.
$$\int \sin ^{3} 2 \theta \cos ^{2} 2 \theta d \theta$$

Check back soon!

Problem 14

Integrate each of the given functions.
$$\int_{0}^{1}\left(1-\cos ^{2} 4 x\right) d x$$

Check back soon!

Problem 15

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

Check back soon!

Problem 16

Integrate each of the given functions.
$$\int \frac{6 \cot ^{2} y}{\tan y} d y$$

Check back soon!

Problem 17

Integrate each of the given functions.
$$\int_{0}^{\pi / 4} \frac{\tan x}{\cos ^{4} x} d x$$

Check back soon!

Problem 18

Integrate each of the given functions.
$$\int \frac{\csc ^{4} 4 x}{\tan 4 x} d x$$

Check back soon!

Problem 19

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

Check back soon!

Problem 20

Integrate each of the given functions.
$$\int 4 \cot ^{4} x d x$$

Check back soon!

Problem 21

Integrate each of the given functions.
$$\int 0.5 \sin s \sin 2 s d s$$

Check back soon!

Problem 22

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

Check back soon!

Problem 23

Integrate each of the given functions.
$$\int(\sin x+\cos x)^{2} d x$$

Check back soon!

Problem 24

Integrate each of the given functions.
$$\int(\tan 2 x+\cot 2 x)^{2} d x$$

Check back soon!

Problem 25

Integrate each of the given functions.
$$\int \frac{1-\cot x}{\sin ^{4} x} d x$$

Check back soon!

Problem 26

Integrate each of the given functions.
$$\int \frac{\left(\sin u+\sin ^{2} u\right)^{2}}{\sec u} d u$$

Check back soon!

Problem 27

Integrate each of the given functions.
$$\int_{\pi / 6}^{\pi / 4} \cot ^{5} p d p$$

Check back soon!

Problem 28

Integrate each of the given functions.
$$\int_{\pi / 6}^{\pi / 3} \frac{2 d x}{1+\sin x}$$

Check back soon!

Problem 29

Integrate each of the given functions.
$$\int \sec ^{6} x d x$$

Check back soon!

Problem 30

Integrate each of the given functions.
$$\int \tan ^{7} x d x$$

Check back soon!

Problem 31

Integrate each of the given functions.
$$\int \frac{\sin 2 x}{\cos ^{3} x} d x$$

Check back soon!

Problem 32

Integrate each of the given functions.
$$\int \frac{\sec ^{2} t \tan t}{4+\sec ^{2} t} d t$$

Check back soon!

Problem 33

Integrate each of the given functions.
$$\int \frac{\sec e^{-x}}{e^{x}} d x$$

Check back soon!

Problem 34

Integrate each of the given functions.
$$\int_{0}^{\pi / 4} \sqrt{1+\cos 4 x} d x$$

Check back soon!

Problem 35

Solve the given problems by integration.
Using the identity $\sin \alpha \cos \beta=\frac{1}{2}[\sin (\alpha+\beta)+\sin (\alpha-\beta)]$
integrate $\int \sin 4 x \cos 5 x d x$

Check back soon!

Problem 36

Solve the given problems by integration.
Using the identity $\cos \alpha \cos \beta=\frac{1}{2}[\cos (\alpha+\beta)+\cos (\alpha-\beta)]$
integrate $\int \cos 3 x \cos 4 x d x$

Check back soon!

Using the identity $\cos \alpha \cos \beta=\frac{1}{2}[\cos (\alpha+\beta)+\cos (\alpha-\beta)]$
integrate $\int \cos 3 x \cos 4 x d x$
Find the volume generated by revolving the region bounded by $y=\sin x$ and $y=0,$ from $x=0$ to $x=\pi,$ about the $x$ -axis.

Check back soon!

Problem 38

Solve the given problems by integration.
Find the volume generated by revolving the region bounded by $y=\tan ^{3}\left(x^{2}\right), y=0,$ and $x=\frac{\pi}{4}$ about the $y$ -axis.

Check back soon!

Problem 39

Solve the given problems by integration.
Find the area bounded by $y=\sin ^{4} x, y=\cos ^{4} x,$ and $x=0$ in the first quadrant. (Hint: Use factoring to simplify the integrand as much as possible.)

Check back soon!

Problem 40

Solve the given problems by integration.
The length of an arc along a function is given by $s=\int_{a}^{b} \sqrt{1+\left(\frac{d y}{d x}\right)^{2}}$ $d x .$ Find the arc length of the curve $y=\ln \cos x$ from $x=0$ to $x=\frac{\pi}{3}$

Check back soon!

Problem 41

Solve the given problems by integration.
Show that $\int \sin x \cos x d x$ can be integrated in two ways. Explain the difference in the answers.

Check back soon!

Problem 42

Solve the given problems by integration.
For $n > 0,$ show that $\int \tan x \sec ^{n} x d x=1 \sec ^{n} x+C$

Check back soon!

Problem 43

Solve the given problems by integration.
Show that $\int_{0}^{\pi} \sin ^{2} n x d x=\frac{1}{2} \pi,$ where $n$ is any positive integer.

Check back soon!

Problem 44

Solve the given problems by integration.
The velocity $v$ (in $\mathrm{cm} / \mathrm{s}$ ) of an object is $v=\cos ^{2} \pi t .$ How far does the object move in $4.0 \mathrm{s} ?$

Check back soon!

Problem 45

Solve the given problems by integration.
The acceleration $a$ (in $\mathrm{f} / \mathrm{s}^{2}$ ) of an object is $a=\sin ^{2} t \cos t$. If the object starts at the origin with a velocity of $6 \mathrm{ft} / \mathrm{s}$, what is its position at time $t ?$

Check back soon!

Problem 46

Solve the given problems by integration.
Find the root-mean-square current in a circuit from $t=0$ s to $t=0.50 \mathrm{s}$ if $i=i_{0} \sin t \sqrt{\cos t}$

Check back soon!

Problem 47

Solve the given problems by integration.
In the study of the rate of radiation by an accelerated charge, the following integral must be evaluated: $\int_{0}^{\pi} \sin ^{3} \theta d \theta$ of the integral.

Check back soon!

Problem 48

Solve the given problems by integration.
In finding the volume of a special O-ring for a space vehicle, the $\int \frac{\sin ^{2} \theta}{\cos ^{2} \theta} d \theta$ must be evaluated. Perform this integration.

Check back soon!

Problem 49

Solve the given problems by integration.
For a voltage $V=340 \sin 120 \pi t,$ show that the root-mean-square voltage for one period is $240 \mathrm{V}$

Check back soon!

Problem 50

Solve the given problems by integration.
For a current $i=i_{0} \sin \omega t,$ show that the root-mean-square current for one period is $i_{0} / \sqrt{2}$

Check back soon!

Problem 51

Solve the given problems by integration.
In the analysis of the intensity of light from a certain source, the equation $I=A \int_{-a / 2}^{a / 2} \cos ^{2}[b \pi(c-x)] d x$ is used. Here, $A, a, b,$ and $c$ are constants. Evaluate this integral. (The simplification is quite lengthy.)

Check back soon!

Problem 52

Solve the given problems by integration.
In the study of the lifting force $L$ due to a stream of fluid passing around a cylinder, the equation $L=k \int_{0}^{2 \pi}\left(a \sin \theta+b \sin ^{2} \theta-b \sin ^{3} \theta\right) d \theta$
is used. Here, $k, a,$ and $b$ are constants and $\theta$ is the angle from the direction of flow. Evaluate the integral.

Check back soon!