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## Educators

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.

### 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.

### Problem 3

Evaluate the integral.

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

DQ
Danjoseph Q.

### Problem 4

Evaluate the integral.

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

SL
Sky L.

### Problem 5

Evaluate the integral.

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

DQ
Danjoseph Q.

### Problem 6

Evaluate the integral.

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

DQ
Danjoseph Q.

### Problem 7

Evaluate the integral.

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

WZ
Wen Z.

### Problem 8

Evaluate the integral.

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

Ma. Theresa A.

### Problem 9

Evaluate the integral.

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

WZ
Wen Z.

### Problem 10

Evaluate the integral.

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

WZ
Wen Z.

### Problem 11

Evaluate the integral.

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

WZ
Wen Z.

### Problem 12

Evaluate the integral.

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

WZ
Wen Z.

### Problem 13

Evaluate the integral.

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

WZ
Wen Z.

### Problem 14

Evaluate the integral.

$\displaystyle \int x \cosh ax dx$

WZ
Wen Z.

### Problem 15

Evaluate the integral. $\int \frac{x-1}{x^{2}+2 x} d x$

Anthony H.

### Problem 16

Evaluate the integral.

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

WZ
Wen Z.

### Problem 17

Evaluate the integral.

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

WZ
Wen Z.

### Problem 18

Evaluate the integral.

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

WZ
Wen Z.

### Problem 19

Evaluate the integral.

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

WZ
Wen Z.

### Problem 20

Evaluate the integral.

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

WZ
Wen Z.

### Problem 21

Evaluate the integral.

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

WZ
Wen Z.

### Problem 22

Evaluate the integral.

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

WZ
Wen Z.

### Problem 23

Evaluate the integral.

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

WZ
Wen Z.

### Problem 24

Evaluate the integral.

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

WZ
Wen Z.

### Problem 25

Evaluate the integral.

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

Willis J.

### Problem 26

Evaluate the integral.

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

Anthony H.

### Problem 27

Evaluate the integral.

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

WZ
Wen Z.

### Problem 28

Evaluate the integral.

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

WZ
Wen Z.

### Problem 29

Evaluate the integral.

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

Ma. Theresa A.

### Problem 30

Evaluate the integral.

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

WZ
Wen Z.

### Problem 31

Evaluate the integral.

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

WZ
Wen Z.

### Problem 32

Evaluate the integral.

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

WZ
Wen Z.

### Problem 33

Evaluate the integral.

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

WZ
Wen Z.

### Problem 34

Evaluate the integral.

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

WZ
Wen Z.

### Problem 35

Evaluate the integral.

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

WZ
Wen Z.

### Problem 36

Evaluate the integral.

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

WZ
Wen Z.

### 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.

### 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.

### 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.

### 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.

### 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.

### 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.

### 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.

### 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.

### 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.

### 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.

### 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 $\displaystyle \int \sin^4 x dx$.

Willis J.

### 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 $\displaystyle \int \cos^4 x dx$.

Carson M.

### 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,
$$\int_0^{\frac{\pi}{2}} \sin^{2n + 1} x dx = \frac{2 \cdot 4 \cdot 6 \cdots \cdots 2n}{3 \cdot 5 \cdot 7 \cdots \cdots (2n +1)}$$

Willis J.

### 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.

### 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.

### 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.

### 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.

### 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.

### Problem 55

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

WZ
Wen Z.

### Problem 56

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

WZ
Wen Z.

### 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.

### Problem 58

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

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

WZ
Wen Z.

### 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.

### 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.

### 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.

### 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.

### 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.

### 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.

### 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.

### Problem 66

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

WZ
Wen Z.

### 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 $S(x)$.]

Carson M.

### 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.

Carson M.

### 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.

### 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.

### 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 $\displaystyle \int_1^4 x f^{\prime\prime} (x)\ dx$.

Ma. Theresa A.

### 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.

### 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.

### 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.