1.13: More Integration Examples

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 \(\ln x\text{.}\)

Stage 1

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

Stage 2

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

Stage 3

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