1.13: More Integration Examples

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Exercises

Recall that we are using $$\log x$$ to denote the logarithm of $$x$$ with base $$e\text{.}$$ In other courses it is often denoted $$\ln x\text{.}$$

1

Match the integration method to a common kind of integrand it's used to antidifferentiate.

 (A) $$u=f(x)$$ substitution (I) a function multiplied by its derivative (B) trigonometric substitution (II) a polynomial times an exponential (C) integration by parts (III) a rational function (D) partial fractions (IV) the square root of a quadratic function
2

Evaluate $$\displaystyle\int_{0}^{\frac{\pi }{2}}\sin^{4}x\cos^{5}x \, d{x}\text{.}$$

3

Evaluate $$\displaystyle\int \sqrt{3-5x^2}\, d{x}\text{.}$$

4

Evaluate $$\displaystyle\int_0^\infty \dfrac{x-1}{e^x}\, d{x}\text{.}$$

5

Evaluate $$\displaystyle\int \frac{-2}{3x^2+4x+1}\, d{x}\text{.}$$

6

Evaluate $$\displaystyle\int_1^2 x^2\log x \, d{x}\text{.}$$

7 (✳)

Evaluate $$\displaystyle\int\frac{x}{x^2-3}\,\, d{x}\text{.}$$

8 (✳)

Evaluate the following integrals.

1. $$\displaystyle\int_0^4\frac{x}{\sqrt{9+x^2}}\,\, d{x}$$
2. $$\displaystyle\int_0^{\pi/2}\cos^3x\ \sin^2x\,\, d{x}$$
3. $$\displaystyle\int_1^{e}x^3\log x\,\, d{x}$$
9 (✳)

Evaluate the following integrals.

1. $$\displaystyle\int_0^{\pi/2} x\sin x\,\, d{x}$$
2. $$\displaystyle\int_0^{\pi/2} \cos^5 x\,\, d{x}$$
10 (✳)

Evaluate the following integrals.

1. $$\displaystyle\int_0^2 xe^x\,\, d{x}$$
2. $$\displaystyle\int_0^1\frac{1}{\sqrt{1+x^2}}\,\, d{x}$$
3. $$\displaystyle\int_3^5\frac{4x}{(x^2-1)(x^2+1)}\,\, d{x}$$
11 (✳)

Calculate the following integrals.

1. $$\displaystyle\int_0^3\sqrt{9-x^2}\,\, d{x}$$
2. $$\displaystyle\int_0^1\log(1+x^2)\,\, d{x}$$
3. $$\displaystyle\int_3^\infty\frac{x}{(x-1)^2(x-2)}\,\, d{x}$$
12

Evaluate $$\displaystyle\int\frac{\sin^4\theta-5\sin^3\theta+4\sin^2\theta+10\sin\theta}{\sin^2\theta-5\sin\theta+6}\cos\theta\, d{\theta}\text{.}$$

13 (✳)

Evaluate the following integrals. Show your work.

1. $$\displaystyle\int_0^{\pi\over 4}\sin^2(2x)\cos^3(2x)\ \, d{x}$$
2. $$\displaystyle\int\big(9+x^2\big)^{-{3\over 2}}\ \, d{x}$$
3. $$\displaystyle\int\frac{\, d{x}}{(x-1)(x^2+1)}$$
4. $$\displaystyle\int x\arctan x\ \, d{x}$$
14 (✳)

Evaluate the following integrals.

1. $$\displaystyle\int_0^{\pi/4}\sin^5(2x)\,\cos(2x)\ \, d{x}$$
2. $$\displaystyle\int\sqrt{4-x^2}\ \, d{x}$$
3. $$\displaystyle\int\frac{x+1}{x^2(x-1)}\ \, d{x}$$
15 (✳)

Calculate the following integrals.

1. $$\displaystyle\int_0^\infty e^{-x} \sin(2x)\,\, d{x}$$
2. $$\displaystyle\int_0^{\sqrt{2}}\frac{1}{(2+x^2)^{3/2}}\,\, d{x}$$
3. $$\displaystyle\int_0^1 x\log(1+x^2)\,\, d{x}$$
4. $$\displaystyle\int_3^\infty\frac{1}{(x-1)^2(x-2)}\,\, d{x}$$
16 (✳)

Evaluate the following integrals.

1. $$\displaystyle\int x\,\log x\ \, d{x}$$
2. $$\displaystyle\int\frac{(x-1)\,\, d{x}}{x^2+4x+5}$$
3. $$\displaystyle\int\frac{\, d{x}}{x^2-4x+3}$$
4. $$\displaystyle\int\frac{x^2\,\, d{x}}{1+x^6}$$
17 (✳)

Evaluate the following integrals.

1. $$\displaystyle\int_0^1\arctan x\ \, d{x}\text{.}$$
2. $$\displaystyle\int\frac{2x-1}{x^2-2x+5}\ \, d{x}\text{.}$$(✳)
18 (✳)
1. Evaluate $${\displaystyle \int\frac{x^2}{(x^3 + 1)^{101}}\,\, d{x}}\text{.}$$
2. Evaluate $$\displaystyle\int \cos^3\!x\ \sin^4\!x\ \, d{x}\text{.}$$
19 (✳)

Evaluate $$\displaystyle\int_{\pi/2}^\pi \frac{\cos x}{\sqrt{\sin x}}\, d{x}\text{.}$$

20 (✳)

Evaluate the following integrals.

1. $$\displaystyle\int \frac{e^x}{(e^x+1)(e^x-3)}\, \, d{x}$$
2. $$\displaystyle\int_2^4 \frac{x^2-4x+4}{\sqrt{12+4x-x^2}}\, \, d{x}$$
21 (✳)

Evaluate these integrals.

1. $$\displaystyle\int\frac{\sin^3x}{\cos^3x} \ \, d{x}$$
2. $$\displaystyle\int_{-2}^{2}\frac{x^4}{x^{10}+16}\ \, d{x}$$
22

Evaluate $$\displaystyle\int x\sqrt{x-1}\, d{x}\text{.}$$

23

Evaluate $$\displaystyle\int \frac{\sqrt{x^2-2}}{x^2}\, d{x}\text{.}$$

You may use that $$\int \sec x\, d{x} = \log|\sec x+\tan x| +C\text{.}$$

24

Evaluate $$\displaystyle\int_0^{\pi/4} \sec^4x\tan^5x\,\, d{x}\text{.}$$

25

Evaluate $$\displaystyle\int \frac{3x^2+4x+6}{(x+1)^3} \, \, d{x}\text{.}$$

26

Evaluate $$\displaystyle\int\frac{1}{x^2+x+1}\,\, d{x}\text{.}$$

27

Evaluate $$\displaystyle\int \sin x \cos x \tan x\, d{x}\text{.}$$

28

Evaluate $$\displaystyle\int \frac{1}{x^3+1}\, d{x}\text{.}$$

29

Evaluate $$\displaystyle\int (3x)^2\arcsin x \, d{x}\text{.}$$

30

Evaluate $$\displaystyle\int_0^{\pi/2}\sqrt{\cos t+1}\ \, d{t}\text{.}$$

31

Evaluate $$\displaystyle\int_{0}^{e} \frac{\log\sqrt{x}}{x}\, d{x}\text{.}$$

32

Evaluate $$\displaystyle\int_{0.1}^{0.2} \frac{\tan x}{\log(\cos x)}\, \, d{x}\text{.}$$

33 (✳)

Evaluate these integrals.

1. $$\displaystyle\int\sin(\log x) \ \, d{x}$$
2. $$\displaystyle\int_0^1\frac{1}{x^2-5x+6}\ \, d{x}$$
34 (✳)

Evaluate (with justification).

1. $$\displaystyle\int_0^3(x+1)\sqrt{9-x^2} \ \, d{x}$$
2. $$\displaystyle\int\frac{4x+8}{(x-2)(x^2+4)}\ \, d{x}$$
3. $$\displaystyle\int_{-\infty}^{+\infty} \frac{1}{e^x+e^{-x}}\ \, d{x}$$
35

Evaluate $$\displaystyle\int \sqrt{\frac{x}{1-x}}\, d{x}\text{.}$$

36

Evaluate $$\displaystyle\int_0^1e^{2x}e^{e^x}\,\, d{x}\text{.}$$

37

Evaluate $$\displaystyle\int\frac{xe^x}{(x+1)^2}\, d{x}\text{.}$$

38

Evaluate $$\displaystyle\int \frac{x\sin x}{\cos^2 x}\,\, d{x}\text{.}$$

You may use that $$\int \sec x\, d{x} = \log|\sec x+\tan x| +C\text{.}$$

39

Evaluate $$\displaystyle\int x(x+a)^n\, d{x}\text{,}$$ where $$a$$ and $$n$$ are constants.

40

Evaluate $$\displaystyle\int\arctan (x^2)\, d{x}\text{.}$$

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