Chapter 12

Determine whether the improper integral diverges or converges. Evaluate the integral if it converges. $$ \int_{1}^{\infty} \frac{1}{\sqrt[3]{x}} d x $$

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

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

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. $$ \int_{0}^{8} \sqrt[3]{x} d x, n=8 $$

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. $$ \int_{0}^{1} \frac{1}{1+x} d x, n=4 $$

Use partial fractions to find the indefinite integral. $$ \int \frac{1}{x^{2}-1} d x $$

Find the indefinite integral. (Hint: Integration by parts is not required for all the integrals.) $$ \int x e^{4 x} d x $$

Determine whether the improper integral diverges or converges. Evaluate the integral if it converges. $$ \int_{0}^{\infty} \frac{5}{e^{2 x}} d x $$

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. $$ \int_{0}^{2} x \sqrt{x^{2}+1} d x, n=4 $$

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

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

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

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

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.) $$ \int_{0}^{1} \frac{1}{1+x^{2}} d x, n=4 $$

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

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

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.) $$ \int_{0}^{2} \frac{1}{\sqrt{1+x^{3}}} d x, n=4 $$

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

Determine whether the improper integral diverges or converges. Evaluate the integral if it converges. $$ \int_{-\infty}^{0} e^{-x} d x $$

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.) $$ \int_{0}^{2} \sqrt{1+x^{3}} d x, n=4 $$

Use partial fractions to find the indefinite integral. $$ \int \frac{1}{2 x^{2}-x} d x $$

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

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

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.) $$ \int_{0}^{1} \sqrt{1-x} d x, n=4 $$

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

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

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

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.) $$ \int_{0}^{1} \sqrt{1-x^{2}} d x, n=4 $$

Use partial fractions to find the indefinite integral. $$ \int \frac{10}{x^{2}-10 x} d x $$

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

Determine whether the improper integral diverges or converges. Evaluate the integral if it converges. $$ \int_{-\infty}^{0} \frac{x}{x^{2}+1} d x $$

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.) $$ \int_{0}^{1} \sqrt{1-x^{2}} d x, n=8 $$

Use partial fractions to find the indefinite integral. $$ \int \frac{5}{x^{2}+x-6} d x $$

Find the indefinite integral. (Hint: Integration by parts is not required for all the integrals.) $$ \int \frac{1}{2} x^{3} e^{x} d x $$

Determine whether the improper integral diverges or converges. Evaluate the integral if it converges. $$ \int_{-\infty}^{\infty} 2 x e^{-3 x^{2}} d x $$

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.) $$ \int_{0}^{2} e^{-x^{2}} d x, n=2 $$

Use partial fractions to find the indefinite integral. $$ \int \frac{3}{x^{2}+x-2} d x $$

Find the indefinite integral. (Hint: Integration by parts is not required for all the integrals.) $$ \int t \ln (t+1) d t $$

Determine whether the improper integral diverges or converges. Evaluate the integral if it converges. $$ \int_{-\infty}^{\infty} x^{2} e^{-x^{3}} d x $$

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.) $$ \int_{0}^{2} e^{-x^{2}} d x, n=4 $$

Use partial fractions to find the indefinite integral. $$ \int \frac{1}{4 x^{2}-9} d x $$

Find the indefinite integral. (Hint: Integration by parts is not required for all the integrals.) $$ \int x^{3} \ln x d x $$

Determine whether the improper integral diverges or converges. Evaluate the integral if it converges, and check your results with the results obtained by using the integration capabilities of a graphing utility. $$ \int_{0}^{1} \frac{1}{1-x} d x $$

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.) $$ \int_{0}^{3} \frac{1}{2-2 x+x^{2}} d x, n=6 $$

Use partial fractions to find the indefinite integral. $$ \frac{5-x}{2 x^{2}+x-1} d x $$

Find the indefinite integral. (Hint: Integration by parts is not required for all the integrals.) $$ \int(x-1) e^{x} d x $$

Determine whether the improper integral diverges or converges. Evaluate the integral if it converges, and check your results with the results obtained by using the integration capabilities of a graphing utility. $$ \int_{0}^{27} \frac{5}{\sqrt[3]{x}} d x $$

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.) $$ \int_{0}^{3} \frac{x}{2+x+x^{2}} d x, n=6 $$

Use partial fractions to find the indefinite integral. $$ \int \frac{x+1}{x^{2}+4 x+3} d x $$