91Ó°ÊÓ

Evaluate the indefinite integrals in Exercises \(1-16\) by using the given substitutions to reduce the integrals to standard form. $$ \int \frac{9 r^{2} d r}{\sqrt{1-r^{3}}}, \quad u=1-r^{3} $$

Suppose that \(\int_{1}^{2} f(x) d x=5 .\) Find $$ \begin{array}{ll}{\text { a. } \int_{1}^{2} f(u) d u} & {\text { b. } \int_{1}^{2} \sqrt{3} f(z) d z} \\ {\text { c. } \int_{2}^{1} f(t) d t} & {\text { d. } \int_{1}^{2}[-f(x)] d x}\end{array} $$

Express the sums in sigma notation. The form of your answer will depend on your choice for the starting index. $$ 1+2+3+4+5+6 $$

Evaluate the integrals $$ \int_{\pi / 4}^{3 \pi / 4} \csc \theta \cot \theta d \theta $$

Express the sums in sigma notation. The form of your answer will depend on your choice for the starting index. $$ 1+4+9+16 $$

Evaluate the indefinite integrals in Exercises \(1-16\) by using the given substitutions to reduce the integrals to standard form. $$ \int 12\left(y^{4}+4 y^{2}+1\right)^{2}\left(y^{3}+2 y\right) d y, \quad u=y^{4}+4 y^{2}+1 $$

Evaluate the integrals $$ \int_{0}^{\pi / 3} 4 \frac{\sin u}{\cos ^{2} u} d u $$

Suppose that \(\int_{-3}^{0} g(t) d t=\sqrt{2} .\) Find $$ \begin{array}{ll}{\text { a. } \int_{0}^{-3} g(t) d t} & {\text { b. } \int_{-3}^{0} g(u) d u} \\ {\text { c. }} {\int_{-3}^{0}[-g(x)] d x} & {\text { d. } \int_{-3}^{0} \frac{g(r)}{\sqrt{2}} d r}\end{array} $$

Evaluate the integrals $$ \int_{\pi / 2}^{0} \frac{1+\cos 2 t}{2} d t $$

Evaluate the indefinite integrals in Exercises \(1-16\) by using the given substitutions to reduce the integrals to standard form. $$ \int \sqrt{x} \sin ^{2}\left(x^{3 / 2}-1\right) d x, \quad u=x^{3 / 2}-1 $$