More Integration Examples

Section 1.13 More Integration Examples

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

Exercises — 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

Exercises — Stage 2 .

2.

Evaluate \(\displaystyle\int_0^{{\pi}/{2}} \sin^4 x \cos^5 x \dee{x}\text{.}\)

3.

Evaluate \(\displaystyle\int \sqrt{3-5x^2}\dee{x}\text{.}\)

4.

Evaluate \(\displaystyle\int_0^\infty \dfrac{x-1}{e^x}\dee{x}\text{.}\)

5.

Evaluate \(\displaystyle\int \frac{-2}{3x^2+4x+1}\dee{x}\text{.}\)

6.

Evaluate \(\displaystyle\int_1^2 x^2\log x \dee{x}\text{.}\)

7. (✳).

Evaluate \(\displaystyle\int\frac{x}{x^2-3}\,\dee{x}\text{.}\)

8. (✳).

Evaluate the following integrals.

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

9. (✳).

Evaluate the following integrals.

\(\displaystyle \displaystyle\int_0^{\pi/2} x\sin x\,\dee{x} \)
\(\displaystyle \displaystyle\int_0^{\pi/2} \cos^5 x\,\dee{x} \)

10. (✳).

Evaluate the following integrals.

\(\displaystyle \displaystyle\int_0^2 xe^x\,\dee{x}\)
\(\displaystyle \displaystyle\int_0^1\frac{1}{\sqrt{1+x^2}}\,\dee{x}\)
\(\displaystyle \displaystyle\int_3^5\frac{4x}{(x^2-1)(x^2+1)}\,\dee{x}\)

11. (✳).

Calculate the following integrals.

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

12.

Evaluate \(\displaystyle\int\frac{\sin^4\theta-5\sin^3\theta+4\sin^2\theta+10\sin\theta}{\sin^2\theta-5\sin\theta+6}\cos\theta\dee{\theta}\text{.}\)

13. (✳).

Evaluate the following integrals. Show your work.

\(\displaystyle \displaystyle\int_0^{\pi\over 4}\sin^2(2x)\cos^3(2x)\ \dee{x}\)
\(\displaystyle \displaystyle\int\big(9+x^2\big)^{-{3\over 2}}\ \dee{x}\)
\(\displaystyle \displaystyle\int\frac{\dee{x}}{(x-1)(x^2+1)}\)
\(\displaystyle \displaystyle\int x\arctan x\ \dee{x}\)

14. (✳).

Evaluate the following integrals.

\(\displaystyle \displaystyle\int_0^{\pi/4}\sin^5(2x)\,\cos(2x)\ \dee{x}\)
\(\displaystyle \displaystyle\int\sqrt{4-x^2}\ \dee{x}\)
\(\displaystyle \displaystyle\int\frac{x+1}{x^2(x-1)}\ \dee{x}\)

15. (✳).

Calculate the following integrals.

\(\displaystyle \displaystyle\int_0^\infty e^{-x} \sin(2x)\,\dee{x}\)
\(\displaystyle \displaystyle\int_0^{\sqrt{2}}\frac{1}{(2+x^2)^{3/2}}\,\dee{x}\)
\(\displaystyle \displaystyle\int_0^1 x\log(1+x^2)\,\dee{x}\)
\(\displaystyle \displaystyle\int_3^\infty\frac{1}{(x-1)^2(x-2)}\,\dee{x}\)

16. (✳).

Evaluate the following integrals.

\(\displaystyle \displaystyle\int x\,\log x\ \dee{x}\)
\(\displaystyle \displaystyle\int\frac{(x-1)\,\dee{x}}{x^2+4x+5}\)
\(\displaystyle \displaystyle\int\frac{\dee{x}}{x^2-4x+3}\)
\(\displaystyle \displaystyle\int\frac{x^2\,\dee{x}}{1+x^6}\)

17. (✳).

Evaluate the following integrals.

\(\displaystyle\int_0^1\arctan x\ \dee{x}\text{.}\)
\(\displaystyle\int\frac{2x-1}{x^2-2x+5}\ \dee{x}\text{.}\)

18. (✳).

Evaluate \({\displaystyle \int\frac{x^2}{(x^3 + 1)^{101}}\,\dee{x}}\text{.}\)
Evaluate \(\displaystyle\int \cos^3\!x\ \sin^4\!x\ \dee{x}\text{.}\)

19.

Evaluate \(\displaystyle\int_{\pi/2}^\pi \frac{\cos x}{\sqrt{\sin x}}\dee{x}\text{.}\)

20. (✳).

Evaluate the following integrals.

\(\displaystyle \displaystyle\int \frac{e^x}{(e^x+1)(e^x-3)}\, \dee{x}\)
\(\displaystyle \displaystyle\int_2^4 \frac{x^2-4x+4}{\sqrt{12+4x-x^2}}\, \dee{x}\)

21. (✳).

Evaluate these integrals.

\(\displaystyle \displaystyle\int\frac{\sin^3x}{\cos^3x} \ \dee{x}\)
\(\displaystyle \displaystyle\int_{-2}^{2}\frac{x^4}{x^{10}+16}\ \dee{x}\)

22.

Evaluate \(\displaystyle\int x\sqrt{x-1}\dee{x}\text{.}\)

23.

Evaluate \(\displaystyle\int \frac{\sqrt{x^2-2}}{x^2}\dee{x}\) for \(x\ge\sqrt{2}\text{.}\)

You may use that \(\int \sec x\dee{x} = \log|\sec x+\tan x| +C\text{.}\)

24.

Evaluate \(\displaystyle\int_0^{\pi/4} \sec^4x\tan^5x\,\dee{x}\text{.}\)

25.

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

26.

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

27.

Evaluate \(\displaystyle\int \sin x \cos x \tan x\dee{x}\text{.}\)

28.

Evaluate \(\displaystyle\int \frac{1}{x^3+1}\dee{x}\text{.}\)

29.

Evaluate \(\displaystyle\int (3x)^2\arcsin x \dee{x}\text{.}\)

Exercises — Stage 3 .

30.

Evaluate \(\displaystyle\int_0^{\pi/2}\sqrt{\cos t+1}\ \dee{t}\text{.}\)

31.

Evaluate \(\displaystyle\int_1^e \frac{\log\sqrt{x}}{{x}}\,\dee{x}\text{.}\)

32.

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

33. (✳).

Evaluate these integrals.

\(\displaystyle \displaystyle\int\sin(\log x) \ \dee{x}\)
\(\displaystyle \displaystyle\int_0^1\frac{1}{x^2-5x+6}\ \dee{x}\)

34. (✳).

Evaluate (with justification).

\(\displaystyle \displaystyle\int_0^3(x+1)\sqrt{9-x^2} \ \dee{x}\)
\(\displaystyle \displaystyle\int\frac{4x+8}{(x-2)(x^2+4)}\ \dee{x}\)
\(\displaystyle \displaystyle\int_{-\infty}^{+\infty} \frac{1}{e^x+e^{-x}}\ \dee{x}\)

35.

Evaluate \(\displaystyle\int \sqrt{\frac{x}{1-x}}\dee{x}\text{.}\)

36.

Evaluate \(\displaystyle\int_0^1e^{2x}e^{e^x}\,\dee{x}\text{.}\)

37.

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

38.

Evaluate \(\displaystyle\int \frac{x\sin x}{\cos^2 x}\,\dee{x}\text{.}\)

You may use that \(\int \sec x\dee{x} = \log|\sec x+\tan x| +C\text{.}\)

39.

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

40.

Evaluate \(\displaystyle\int\arctan (x^2)\dee{x}\text{.}\)