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Use the Substitution Rule for Definite Integrals to evaluate each definite integral. $$ \int_{0}^{\pi / 2} \cos ^{2} x \sin x d x $$

Use symmetry to help you evaluate the given integral. $$ \int_{-\pi / 4}^{\pi / 4}\left(|x| \sin ^{5} x+|x|^{2} \tan x\right) d x $$

Use the Substitution Rule for Definite Integrals to evaluate each definite integral. $$ \int_{0}^{\pi / 2} \sin ^{2} 3 x \cos 3 x d x $$

Use the Substitution Rule for Definite Integrals to evaluate each definite integral. $$ \int_{0}^{1}(x+1)\left(x^{2}+2 x\right)^{2} d x $$

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?

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

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 the Substitution Rule for Definite Integrals to evaluate each definite integral. $$ \int_{1}^{4} \frac{(\sqrt{x}-1)^{3}}{\sqrt{x}} d x $$

Use the Substitution Rule for Definite Integrals to evaluate each definite integral. $$ \int_{0}^{\pi / 6} \sin ^{3} \theta \cos \theta d \theta $$