###### 2

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

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

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

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

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

###### 6

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

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

######
7(*)

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

HintThe 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.

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

HintYou 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.

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

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

######
10(*)

Evaluate the following integrals.

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

HintFor 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.

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

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

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

######
13(*)

Evaluate the following integrals. *Show your work.*

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

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

######
14(*)

Evaluate the following integrals.

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

######
15(*)

Calculate the following integrals.

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

######
16(*)

Evaluate the following integrals.

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

HintFor 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.

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

HintFor 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(*)

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

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

###### 19

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

HintThe integral is improper.

######
20(*)

Evaluate the following integrals.

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

HintFor 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.

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

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

###### 22

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

###### 23

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

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

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

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

###### 25

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

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

###### 26

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

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

###### 27

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

###### 28

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

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

###### 29

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

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