Books(current) Courses (current) Earn 💰 Log in(current)

Chapter 7

Techniques of Integration

Educators

SL
DQ
WZ

Problem 1

Evaluate the integral using integration by parts with the indicated choices of $ u $ and $ dv $.

$ \displaystyle \int xe^{2x} $ ; $ u = x $ , $ dv = e^{2x} dx $

SL
Sky L.
Numerade Educator

Problem 2

Evaluate the integral using integration by parts with the indicated choices of $ u $ and $ dv $.

$ \displaystyle \int \sqrt{x} \ln x dx $ ; $ u = \displaystyle \ln x $ , $ dv = \sqrt{x} dx $

DQ
Danjoseph Q.
Numerade Educator

Problem 3

Evaluate the integral.

$ \displaystyle \int x \cos 5x dx $

DQ
Danjoseph Q.
Numerade Educator

Problem 4

Evaluate the integral.

$ \displaystyle \int ye^{0.2y} dy $

SL
Sky L.
Numerade Educator

Problem 5

Evaluate the integral.

$ \displaystyle \int te^{-3t} dt $

DQ
Danjoseph Q.
Numerade Educator

Problem 6

Evaluate the integral.

$ \displaystyle \int (x - 1) \sin \pi x dx $

DQ
Danjoseph Q.
Numerade Educator

Problem 7

Evaluate the integral.

$ \displaystyle \int (x^2 + 2x) \cos x dx $

WZ
Wen Z.
Numerade Educator

Problem 8

Evaluate the integral.

$ \displaystyle \int t^2 \sin \beta t dt $

WZ
Wen Z.
Numerade Educator

Problem 9

Evaluate the integral.

$ \displaystyle \int \cos^{-1} x dx $

WZ
Wen Z.
Numerade Educator

Problem 10

Evaluate the integral.

$ \displaystyle \int \ln \sqrt{x} dx $

WZ
Wen Z.
Numerade Educator

Problem 11

Evaluate the integral.

$ \displaystyle \int t^4 \ln t dt $

WZ
Wen Z.
Numerade Educator

Problem 12

Evaluate the integral.

$ \displaystyle \int \tan^{-1} 2y dy $

WZ
Wen Z.
Numerade Educator

Problem 13

Evaluate the integral.

$ \displaystyle \int t \csc^2 t dt $

WZ
Wen Z.
Numerade Educator

Problem 14

Evaluate the integral.

$ \displaystyle \int x \cosh ax dx $

WZ
Wen Z.
Numerade Educator

Problem 15

Evaluate the integral.

$ \displaystyle \int (\ln x)^2 dx $

WZ
Wen Z.
Numerade Educator

Problem 16

Evaluate the integral.

$ \displaystyle \int \frac{z}{10^z} dz $

WZ
Wen Z.
Numerade Educator

Problem 17

Evaluate the integral.

$ \displaystyle \int e^{2 \theta} \sin 3 \theta d \theta $

WZ
Wen Z.
Numerade Educator

Problem 18

Evaluate the integral.

$ \displaystyle \int e^{-\theta} \cos 2 \theta d \theta $

WZ
Wen Z.
Numerade Educator

Problem 19

Evaluate the integral.

$ \displaystyle \int z^3 e^z dz $

WZ
Wen Z.
Numerade Educator

Problem 20

Evaluate the integral.

$ \displaystyle \int x \tan^2 x dx $

WZ
Wen Z.
Numerade Educator

Problem 21

Evaluate the integral.

$ \displaystyle \int \frac{xe^{2x}}{(1 + 2x)^2} dx $

WZ
Wen Z.
Numerade Educator

Problem 22

Evaluate the integral.

$ \displaystyle \int (\arcsin x)^2 dx $

WZ
Wen Z.
Numerade Educator

Problem 23

Evaluate the integral.

$ \displaystyle \int_0^{\frac{1}{2}} x \cos \pi x dx $

WZ
Wen Z.
Numerade Educator

Problem 24

Evaluate the integral.

$ \displaystyle \int_0^1 (x^2 + 1) e^{-x} dx $

WZ
Wen Z.
Numerade Educator

Problem 25

Evaluate the integral.

$ \displaystyle \int_0^2 y \sinh y dy $

WZ
Wen Z.
Numerade Educator

Problem 26

Evaluate the integral.

$ \displaystyle \int_1^2 w^2 \ln w dw $

WZ
Wen Z.
Numerade Educator

Problem 27

Evaluate the integral.

$ \displaystyle \int_1^5 \frac{\ln R}{R^2} dR $

WZ
Wen Z.
Numerade Educator

Problem 28

Evaluate the integral.

$ \displaystyle \int_0^{2 \pi} t^2 \sin 2t dt $

WZ
Wen Z.
Numerade Educator

Problem 29

Evaluate the integral.

$ \displaystyle \int_0^\pi x \sin x \cos x dx $

WZ
Wen Z.
Numerade Educator

Problem 30

Evaluate the integral.

$ \displaystyle \int_1^{\sqrt{3}} \arctan (\frac{1}{x}) dx $

WZ
Wen Z.
Numerade Educator

Problem 31

Evaluate the integral.

$ \displaystyle \int_1^5 \frac{M}{e^M} dM $

WZ
Wen Z.
Numerade Educator

Problem 32

Evaluate the integral.

$ \displaystyle \int_1^2 \frac{(\ln x)^2}{x^3} dx $

WZ
Wen Z.
Numerade Educator

Problem 33

Evaluate the integral.

$ \displaystyle \int_0^{\frac{\pi}{3}} \sin x \ln (\cos x) dx $

WZ
Wen Z.
Numerade Educator

Problem 34

Evaluate the integral.

$ \displaystyle \int_0^1 \frac{r^3}{\sqrt{4 + r^2}} dr $

WZ
Wen Z.
Numerade Educator

Problem 35

Evaluate the integral.

$ \displaystyle \int_1^2 x^4 (\ln x)^2 dx $

WZ
Wen Z.
Numerade Educator

Problem 36

Evaluate the integral.

$ \displaystyle \int_0^t e^s \sin (t - s) ds $

WZ
Wen Z.
Numerade Educator

Problem 37

First make a substitution and then use integration by parts to evaluate the integral.

$ \displaystyle \int e^{\sqrt{x}} dx $

WZ
Wen Z.
Numerade Educator

Problem 38

First make a substitution and then use integration by parts to evaluate the integral.

$ \displaystyle \int \cos (\ln x) dx $

WZ
Wen Z.
Numerade Educator

Problem 39

First make a substitution and then use integration by parts to evaluate the integral.

$ \displaystyle \int_{\sqrt{\frac{\pi}{2}}}^{\sqrt{\pi}} \theta^3 \cos (\theta^2) d \theta $

WZ
Wen Z.
Numerade Educator

Problem 40

First make a substitution and then use integration by parts to evaluate the integral.

$ \displaystyle \int_0^\pi e^{\cos t} \sin 2t dt $

WZ
Wen Z.
Numerade Educator

Problem 41

First make a substitution and then use integration by parts to evaluate the integral.

$ \displaystyle \int x \ln (1 + x) dx $

WZ
Wen Z.
Numerade Educator

Problem 42

First make a substitution and then use integration by parts to evaluate the integral.

$ \displaystyle \int \frac{\arcsin (\ln x)}{x} dx $

WZ
Wen Z.
Numerade Educator

Problem 43

Evaluate the indefinite integral. Illustrate, and check that your answer is reasonable, by graphing both the function and its antiderivative (take $ C = 0 $).

$ \displaystyle \int xe^{-2x} dx $

WZ
Wen Z.
Numerade Educator

Problem 44

Evaluate the indefinite integral. Illustrate, and check that your answer is reasonable, by graphing both the function and its antiderivative (take $ C = 0 $).

$ \displaystyle \int x^{\frac{3}{2}} \ln x dx $

WZ
Wen Z.
Numerade Educator

Problem 45

Evaluate the indefinite integral. Illustrate, and check that your answer is reasonable, by graphing both the function and its antiderivative (take $ C = 0 $).

$ \displaystyle \int x^3 \sqrt{1 + x^2} dx $

WZ
Wen Z.
Numerade Educator

Problem 46

Evaluate the indefinite integral. Illustrate, and check that your answer is reasonable, by graphing both the function and its antiderivative (take $ C = 0 $).

$ \displaystyle \int x^2 \sin 2x dx $

WZ
Wen Z.
Numerade Educator

Problem 47

(a) Use the reduction formula in Example 6 to show that
$$ \int \sin^2 x dx = \frac{x}{2} - \frac{\sin 2x}{4} + C $$
(b) Use part (a) and the reduction formula to evaluate $ \displaystyle \int \sin^4 x dx $.

WZ
Wen Z.
Numerade Educator

Problem 48

(a) Prove the reduction formula
$$ \int \cos^n x dx = \frac{1}{n} \cos^{n - 1} x \sin x + \frac{n - 1}{n} \int \cos^{n - 2} x dx $$
(b) Use part (a) to evaluate $ \displaystyle \int \cos^2 x dx $.
(c) Use parts (a) and (b) to evaluate $ \displaystyle \int \cos^4 x dx $.

WZ
Wen Z.
Numerade Educator

Problem 49

(a) Use the reduction formula in Example 6 to show that
$$ \int_0^{\frac{\pi}{2}} \sin^n x dx = \frac{n - 1}{n} \int_0^{\frac{\pi}{2}} \sin^{n - 2} x dx $$
where $ n \ge 2 $ is an integer.
(b) Use part (a) to evaluate $ \displaystyle \int_0^{\frac{\pi}{2}} \sin^3 x dx $ and $ \displaystyle \int_0^{\frac{\pi}{2}} \sin^5 x dx $.
(c) Use part (a) to show that, for odd powers of sine,
$$ \int_0^{\frac{\pi}{2}} \sin^{2n + 1} x dx = \frac{2 \cdot 4 \cdot 6 \cdots \cdots 2n}{3 \cdot 5 \cdot 7 \cdots \cdots (2n +1)} $$

WZ
Wen Z.
Numerade Educator

Problem 50

Prove that, for even powers of sine,
$$ \int_0^{\frac{\pi}{2}} \sin^{2n} x dx = \frac{1 \cdot 3 \cdot 5 \cdots \cdots (2n - 1)}{2 \cdot 4 \cdot 6 \cdots \cdots 2n} \frac{\pi}{2} $$

WZ
Wen Z.
Numerade Educator

Problem 51

Use integration by parts to prove the reduction formula.

$ \displaystyle \int (\ln x)^n dx = x (\ln x)^n - n \displaystyle \int (\ln x)^{n - 1} dx $

WZ
Wen Z.
Numerade Educator

Problem 52

Use integration by parts to prove the reduction formula.

$ \displaystyle \int x^n e^x dx = x^n e^x - n \displaystyle \int x^{n - 1} e^x dx $

WZ
Wen Z.
Numerade Educator

Problem 53

Use integration by parts to prove the reduction formula.

$ \displaystyle \int \tan^n x dx = \frac{\tan^{n -1} x}{n - 1} - \displaystyle \int \tan^{n -2} x dx (n \neq 1) $

WZ
Wen Z.
Numerade Educator

Problem 54

Use integration by parts to prove the reduction formula.

$ \displaystyle \int \sec^n x dx = \frac{\tan x \sec^{n - 2} x}{n - 1} + \frac{n - 2}{n - 1} \displaystyle \int \sec^{n - 2} x dx (n \neq 1) $

WZ
Wen Z.
Numerade Educator

Problem 55

Use Exercise 51 to find $ \displaystyle \int (\ln x)^3 dx $.

WZ
Wen Z.
Numerade Educator

Problem 56

Use Exercise 52 to find $ \displaystyle \int x^4 e^x dx $.

WZ
Wen Z.
Numerade Educator

Problem 57

Find the area of the region bounded by the given curves.

$ y = x^2 \ln x $ , $ y = 4 \ln x $

WZ
Wen Z.
Numerade Educator

Problem 58

Find the area of the region bounded by the given curves.

$ y = x^2 e^{-x} $ , $ y = xe^{-x} $

WZ
Wen Z.
Numerade Educator

Problem 59

Use a graph to find approximate x-coordinates of the points of intersection of the given curves. Then find (approximately) the area of the region bounded by the curves.

$ y = \arcsin \left(\frac{1}{2} x \right) $, $ y = 2 - x^2 $

WZ
Wen Z.
Numerade Educator

Problem 60

Use a graph to find approximate x-coordinates of the points of intersection of the given curves. Then find (approximately) the area of the region bounded by the curves.

$ y = x \ln (x + 1) $ , $ y = 3x - x^2 $

WZ
Wen Z.
Numerade Educator

Problem 61

Use the method of cylindrical shells to find the volume generated by rotating the region bounded by the curves about the given axis.

$ y = \cos (\frac{\pi x}{2}) $ , $ y = 0 $ , $ 0 \le x \le 1 $ ; about the y-axis

WZ
Wen Z.
Numerade Educator

Problem 62

Use the method of cylindrical shells to find the volume generated by rotating the region bounded by the curves about the given axis.

$ y = e^x $ , $ y = e^{-x} $ , $ x = 1 $ ; about the y-axis

WZ
Wen Z.
Numerade Educator

Problem 63

Use the method of cylindrical shells to find the volume generated by rotating the region bounded by the curves about the given axis.

$ y = e^{-x} $ , $ y = 0 $ , $ x = -1 $ , $ x = 0 $ ; about $ x = 1 $

WZ
Wen Z.
Numerade Educator

Problem 64

Use the method of cylindrical shells to find the volume generated by rotating the region bounded by the curves about the given axis.

$ y = e^x $ , $ x = 0 $ , $ y = 3 $ ; about the x-axis

WZ
Wen Z.
Numerade Educator

Problem 65

Calculate the volume generated by rotating the region bounded by the curves $ y = \ln x $, $ y = 0 $ and $ x = 2 $ about each axis.
(a) The y-axis
(b) The x-axis

WZ
Wen Z.
Numerade Educator

Problem 66

Calculate the average value of $ f(x) = x \sec^2 x $ on the interval $ [0, \frac{\pi}{4}] $.

WZ
Wen Z.
Numerade Educator

Problem 67

The Fresnel function $ S(x) = \displaystyle \int_0^x \sin \left(\frac{1}{2} \pi t^2 \right) dt $ was discussed in Example 5.3.3 and is used extensively in the theory of optics. Find $ S(x) dx $. [Your answer will involve $ S(x) $.]

WZ
Wen Z.
Numerade Educator

Problem 68

A rocket accelerates by burning its onboard fuel, so its mass decreases with time. Suppose the initial mass of the rocket at liftoff (including its fuel) is $ m $, the fuel is consumed at rate $ r $, and the exhaust gases are ejected with constant velocity $ v_e $ (relative to the rocket). A model for the velocity of the rocket at time $ t $ is given by the equation
$$ v(t) = -gt - v_e \ln \frac{m - rt}{m} $$
where $ g $ is the acceleration due to gravity and t is not too large. If $ g = 9.8 m/s^2 $, $ m = 30,000 kg $, $ r = 160 kg/s $, and $ v_e = 3000 m/s $, find the height of the rocket one minute after liftoff.

WZ
Wen Z.
Numerade Educator

Problem 69

A particle that moves along a straight line has velocity $ v(t) = t^2 e^{-t} $ meters per second after $ t $ seconds. How far will it travel during the first $ t $ seconds?

WZ
Wen Z.
Numerade Educator

Problem 70

If $ f(0) = g(0) = 0 $ and $ f^n $ and $ g^n $ are continuous, show that
$$ \int_0^a f(x) g^{\prime\prime} (x) dx = f(a)g^\prime(a) - f^\prime(a)g(a) + \int_0^a f^{\prime\prime} (x) g(x) dx $$

WZ
Wen Z.
Numerade Educator

Problem 71

Suppose that $ f(1) = 2 $, $ f(4) = 7 $, $ f^\prime(1) = 5 $, $ f^\prime(4) = 3 $ and $ f^{\prime\prime} $ is continuous. Find the value of $ \displaystyle \int_1^4 x f^{\prime\prime} (x)\ dx $.

WZ
Wen Z.
Numerade Educator

Problem 72

(a) Use integration by parts to show that
$$ \int f(x) dx = xf (x) - \int xf^\prime (x) dx $$
(b) If $ f $ and $ g $ are inverse functions and $ f^\prime $ is continuous, prove that
$$ \int_a^b f(x) dx = bf (b) - af (a) - \int_{f(a)}^{f(b)} g(y) dy $$
[Hint: Use part (a) and make the substitution $ y = f(x) $.]
(c) In the case where $ f $ and $ g $ are positive functions and $ b > a > 0 $, draw a diagram to give a geometric interpretation of part (b).
(d) Use part (b) to evaluate $ \displaystyle \int_1^e \ln x dx $.

WZ
Wen Z.
Numerade Educator

Problem 73

We arrived at Formula 6.3.2, $ V = \displaystyle \int_a^b 2 \pi x f(x) dx $, by using cylindrical shells, but now we can use integration by parts to prove it using the slicing method of Section 6.2, at least for the case where $ f $ is one-to-one and therefore has an inverse function $ g $. Use the figure to show that
$$ V = \pi b^2 d - \pi a^2 c - \int_c^d \pi [g(y)]^2 dy $$
Make the substitution $ y = f(x) $ and then use integration by parts on the resulting integral to prove that
$$ V = \int_a^b 2 \pi x f(x) dx $$

WZ
Wen Z.
Numerade Educator

Problem 74

Let $ I_n = \displaystyle \int_0^{\frac{\pi}{2}} \sin^n x dx $.
(a) Show that $ I_{2n + 2} \le I_{2n + 1} \le I_{2n} $.
(b) Use Exercise 50 to show that
$$ \frac{I_{2n + 2}}{I_{2n}} = \frac{2n + 1}{2n + 2} $$
(c) Use parts (a) and (b) to show that
$$ \frac{2n + 1}{2n + 2} \le \frac{I_{2n + 1}}{I_{2n}} \le 1 $$
and deduce that $ \lim_{n\to\infty}\frac{I_{2n + 1}}{I_{2n}} = 1 $.
(d) Use part (c) and Exercises 49 and 50 to show that
$$ \lim_{n\to\infty} \frac{2}{1} \cdot \frac{2}{3} \cdot \frac{4}{3} \cdot \frac{4}{5} \cdot \frac{6}{5} \cdot \frac{6}{7} \cdots \cdots \frac{2n}{2n - 1} \cdot \frac{2n}{2n + 1} = \frac{\pi}{2} $$
This formula is usually written as an infinite product:
$$ \frac{\pi}{2} = \frac{2}{1} \cdot \frac{2}{3} \cdot \frac{4}{3} \cdot \frac{4}{5} \cdot \frac{6}{5} \cdot \frac{6}{7} \cdots $$
and is called the Wallis product.
(e) We construct rectangles as follows. Start with a square of area 1 and attach rectangles of area 1 alternately beside or on top of the previous rectangle (see the figure). Find the limit of the ratios of width to height of these rectangles.

WZ
Wen Z.
Numerade Educator