91Ó°ÊÓ

Evaluate the integral. $$ \int(x+\sin x)^{2} d x $$

Use integration by parts to prove the reduction formula. $$ \int \sec ^{n} x d x=\frac{\tan x \sec ^{n-2} x}{n-1}+\frac{n-2}{n-1} \int \sec ^{n-2} x d x \quad(n \neq 1) $$

Find the average value of the function \(f(x)=\sin ^{2} x \cos ^{3} x\) on the interval \([-\pi, \pi] .\)

$$ \begin{array}{l}{\text { Use a graph of } f(x)=1 /\left(x^{2}-2 x-3\right) \text { to decide whether }} \\ {\int_{0}^{2} f(x) d x \text { is positive or negative. Use the graph to give a }} \\ {\text { rough estimate of the value of the integral and then use partial }} \\ {\text { fractions to find the exact value. }}\end{array} $$

The integral $$ \int_{0}^{\infty} \frac{1}{\sqrt{x}(1+x)} d x $$ is improper for two reasons: The interval \([0, \infty)\) is infinite and the integrand has an infinite discontinuity at \(0 .\) Evaluate it by expressing it as a sum of improper integrals of Type 2 and Type 1 as follows: $$ \int_{0}^{\infty} \frac{1}{\sqrt{x}(1+x)} d x=\int_{0}^{1} \frac{1}{\sqrt{x}(1+x)} d x+\int_{1}^{\infty} \frac{1}{\sqrt{x}(1+x)} d x $$

Evaluate the integral. $$ \int \frac{d x}{x+x \sqrt{x}} $$

Evaluate $$ \int \frac{1}{x^{2}+k} d x $$ by considering several cases for the constant k.

Evaluate the integral. $$ \int \frac{d x}{\sqrt{x}+x \sqrt{x}} $$

$$ \begin{array}{l}{\text { Evaluate } \int \sin x \cos x d x \text { by four methods: }} \\ {\text { (a) the substitution } u=\cos x} \\ {\text { (b) the substitution } u=\sin x} \\ {\text { (c) the identity sin } 2 x=2 \sin x \cos x} \\ {\text { (d) integration by parts }} \\ {\text { Explain the different appearances of the answers. }}\end{array}$$

Find the area of the region bounded by the given curves. $$ y=\sin ^{2} x, \quad y=\sin ^{3} x, \quad 0 \leq x \leq \pi $$