91Ó°ÊÓ

Find the indefinite integral. (Hint: Integration by parts is not required for all the integrals.) $$ \int x e^{-2 x} d x $$

Approximate the integral using (a) the Trapezoidal Rule and (b) Simpson's Rule for the indicated value of \(n\). (Round your answers to three significant digits.) $$ \int_{0}^{1} \frac{1}{1+x^{2}} d x, n=4 $$

$$ \int_{1 / 2}^{\infty} \frac{1}{\sqrt{2 x-1}} d x $$

Find the indefinite integral. (Hint: Integration by parts is not required for all the integrals.) $$ \int x e^{x^{2}} d x $$

Use partial fractions to find the indefinite integral. $$ \int \frac{-2}{x^{2}-16} d x $$

Determine whether the improper integral diverges or converges. Evaluate the integral if it converges. $$ \int_{5}^{\infty} \frac{x}{\sqrt{x^{2}-16}} d x $$

Use partial fractions to find the indefinite integral. $$ \int \frac{-4}{x^{2}-4} d x $$

Approximate the integral using (a) the Trapezoidal Rule and (b) Simpson's Rule for the indicated value of \(n\). (Round your answers to three significant digits.) $$ \int_{0}^{2} \frac{1}{\sqrt{1+x^{3}}} d x, n=4 $$

Determine whether the improper integral diverges or converges. Evaluate the integral if it converges. $$ \int_{1 / 2}^{\infty} \frac{1}{\sqrt{2 x-1}} d x $$

Approximate the integral using (a) the Trapezoidal Rule and (b) Simpson's Rule for the indicated value of \(n\). (Round your answers to three significant digits.) $$ \int_{0}^{2} \sqrt{1+x^{3}} d x, n=4 $$