91Ó°ÊÓ

In Problems 35-62, use the Substitution Rule for Definite Integrals to evaluate each definite integral. \(\int_{0}^{1} x e^{x^{2}} d x\)

In Problems 43-48, use a graphing calculator to graph each integrand. Then use the Boundedness Property (Theorem \(C\) ) to find a lower bound and an upper bound for each definite integral. $$ \int_{1}^{5}\left(3+\frac{2}{x}\right) d x $$

How does \(\int_{-b}^{-a} f(x) d x\) compare with \(\int_{a}^{b} f(x) d x\) when \(f\) is an even function? An odd function?

In Problems 35-62, use the Substitution Rule for Definite Integrals to evaluate each definite integral. \(\int_{1}^{4} \frac{(\sqrt{x}-1)^{3}}{\sqrt{x}} d x\)

Prove (by a substitution) that $$ \int_{a}^{b} f(-x) d x=\int_{-b}^{-a} f(x) d x $$

In Problems 43-48, use a graphing calculator to graph each integrand. Then use the Boundedness Property (Theorem \(C\) ) to find a lower bound and an upper bound for each definite integral. $$ \int_{10}^{20}\left(1+\frac{1}{x}\right)^{5} d x $$

Use periodicity to calculate \(\int_{0}^{4 \pi}|\cos x| d x\).

In Problems 43-48, use a graphing calculator to graph each integrand. Then use the Boundedness Property (Theorem \(C\) ) to find a lower bound and an upper bound for each definite integral. $$ \int_{4 \pi}^{8 \pi}\left(5+\frac{1}{20} \sin ^{2} x\right) d x $$

In Problems 35-62, use the Substitution Rule for Definite Integrals to evaluate each definite integral. \(\int_{0}^{\pi / 6} \sin ^{3} \theta \cos \theta d \theta\)

In Problems 43-48, use a graphing calculator to graph each integrand. Then use the Boundedness Property (Theorem \(C\) ) to find a lower bound and an upper bound for each definite integral. $$ \int_{0.2}^{0.4}\left(0.002+0.0001 \cos ^{2} x\right) d x $$