91Ó°ÊÓ

Write the partial fraction decomposition for the expression. $$ \frac{4 x-13}{x^{2}-3 x-10} $$

Use the indicated formula from the table of integrals in this section to find the indefinite integral. $$ \int \frac{2 x}{\sqrt{x^{4}-9}} d x, \text { Formula } 25 $$

Explain why the integral is improper and determine whether it diverges or converges. Evaluate the integral if it converses. $$ \int_{0}^{4} \frac{1}{\sqrt{x}} d x $$

Integration by parts to find the indefinite integral. $$ \int x e^{3 x} d x $$

Explain why the integral is improper and determine whether it diverges or converges. Evaluate the integral if it converses. $$ \int_{3}^{4} \frac{1}{\sqrt{x-3}} d x $$

Write the partial fraction decomposition for the expression. $$ \frac{7 x+5}{6\left(2 x^{2}+3 x+1\right)} $$

Use the indicated formula from the table of integrals in this section to find the indefinite integral. $$ \int x^{2} \sqrt{x^{2}+9} d x, \text { Formula } 22 $$

Use the Trapezoidal Rule and Simpson's Rule to approximate the value of the definite integral for the indicated value of \(n\). Compare these results with the exact value of the definite integral. Round your answers to four decimal places. $$ \int_{1}^{3}\left(4-x^{2}\right) d x, n=4 $$

Integration by parts to find the indefinite integral. $$ \int x e^{-x} d x $$

Explain why the integral is improper and determine whether it diverges or converges. Evaluate the integral if it converses. $$ \int_{0}^{2} \frac{1}{(x-1)^{2 / 3}} d x $$