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Section1.13More Integration Examples

Subsection1.13.1Exercises

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) \(u\)-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
Hint

Each option in each column should be used exactly once.

Exercises — Stage 2

2

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

Hint

The integrand is the product of sines and cosines. See how this was handled with a substitution in Section 1.8.1 of the CLP-II text.

After your substitution, you should have a polynomial expression in \(u\)—but it might take some simplification to get it into a form you can easily integrate.

3

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

Hint

We notice that the integrand has a quadratic polynomial under the square root. If that polynomial were a perfect square, we could get rid of the square root: try a trig substitution, as in Section 1.9 of the CLP-II text.

The identity \(\sin(2\theta)=2\sin\theta\cos\theta\) might come in handy.

4

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

Hint

Notice the integral is improper. When you compute the limit, l'H\^{o}pital's rule might help.

If you're struggling to think of how to antidifferentiate, try writing \(\dfrac{x-1}{e^x} = (x-1)e^{-x}\text{.}\)

5

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

Hint

Which method usually works for rational functions (the quotient of two polynomials)?

6

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

Hint

It would be nice to replace logarithm with its derivative, \(\dfrac{1}{x}\text{.}\)

7(*)

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

Hint

The integrand is a rational function, so it is possible to use partial fractions. But there is a much easier way!

8(*)

Evaluate the following integrals.

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

You should prepare your own personal internal list of integration techniques ordered from easiest to hardest. You should have associated to each technique your own personal list of signals that you use to decide when the technique is likely to be useful.

9(*)

Evaluate the following integrals.

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

Despite both containing a trig function, the two integrals are easiest to evaluate using different methods.

10(*)

Evaluate the following integrals.

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

For the integral of secant, see See Section 1.8.3 or Example 1.10.5 in the

CLP-II text.

In (c), notice the denominator is not yet entirely factored.

11(*)

Calculate the following integrals.

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

Part (a) can be done by inspection — use a little highschool geometry! Part (b) is reminiscent of the antiderivative of logarithm—how did we find that one out? Part (c) is an improper integral.

12

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

Hint

Use the substitution \(u=\sin\theta\text{.}\)

13(*)

Evaluate the following integrals. Show your work.

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

For (c), try a little algebra to split the integral into pieces that are easy to antidifferentiate.

14(*)

Evaluate the following integrals.

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

If you're stumped, review Sections 1.8, 1.9, and 1.10.

15(*)

Calculate the following integrals.

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

For part (a), see Example 1.7.11. For part (d), see Example 1.10.4.

16(*)

Evaluate the following integrals.

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

For part (b), first complete the square in the denominator. You can save some work by first comparing the derivative of the denominator with the numerator. For part (d) use a simple substitution.

17(*)

Evaluate the following integrals.

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

For part (b), complete the square in the denominator. You can save some work by first comparing the derivative of the denominator with the numerator.

18(*)

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

For part (a), the numerator is the derivative of a function that appears in the denominator.

19

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

Hint

The integral is improper.

20(*)

Evaluate the following integrals.

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

For part (a), can you convert this into a partial fractions integral? For part (b), start by completing the square inside the square root.

21(*)

Evaluate these integrals.

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

For part (b), the numerator is the derivative of a function that is embedded in the denominator.

22

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

Hint

Try a substitution.

23

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

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

Hint

Note the quadratic function under the square root: you can solve this with trigonometric substitution, as in Section 1.9 of the CLP-II text.

24

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

Hint

Try a \(u\)-substitution, as in Section 1.8.2 of the CLP-II text

25

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

Hint

What's the usual trick for evaluating a rational function (quotient of polynomials)?

26

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

Hint

If the denominator were \(x^2+1\text{,}\) the antiderivative would be arctangent.

27

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

Hint

Simplify first.

28

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

Hint

\(x^3+1 = (x+1)(x^2-x+1)\)

29

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

Hint

You have the product of two quite dissimilar functions in the integrand—try integration by parts.

Exercises — Stage 3

30

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

Hint

Use the identity \(\cos(2x) = 2\cos^2x -1\text{.}\)

31

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

Hint

Using logarithm rules can make the integrand simpler.

32

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

Hint

What is the derivative of the function in the denominator? How could that be useful to you?

33(*)

Evaluate these integrals.

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

For part (a), the substitution \(u=\log x\) gives an integral that you have seen before.

34(*)

Evaluate (with justification).

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

For part (a), split the integral in two. One part may be evaluated by interpreting it geometrically, without doing any integration at all. For part (c), multiply both the numerator and denominator by \(e^x\) and then make a substitution.

35

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

Hint

Let \(u=\sqrt{1-x}\text{.}\)

36

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

Hint

Use the substitution \(u=e^x\text{.}\)

37

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

Hint

Use integration by parts. If you choose your parts well, the resulting integration will be very simple.

38

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

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

Hint

\(\frac{\sin x}{\cos^2 x} = \tan x \sec x\)

39

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

Hint

The cases \(n=-1\) and \(n=-2\) are different from all other values of \(n\text{.}\)

40

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

Hint

\(x^4+1 = (x^2+\sqrt2x+1)(x^2-\sqrt2x+1)\)