Chapter 12

Decide whether the integral is improper. Explain your reasoning. $$ \int_{0}^{1} \frac{d x}{3 x-2} $$

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^{2} d x, n=4

Use the indicated formula from the table of integrals in this section to find the indefinite integral. $$ \int \frac{x}{(2+3 x)^{2}} d x, \text { Formula } 4 $$

Write the partial fraction decomposition for the expression. $$ \frac{2(x+20)}{x^{2}-25} $$

Decide whether the integral is improper. Explain your reasoning. $$ \int_{1}^{3} \frac{d x}{x^{2}} $$

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

Use the indicated formula from the table of integrals in this section to find the indefinite integral. $$ \int \frac{1}{x(2+3 x)^{2}} d x, \text { Formula } 11 $$

Write the partial fraction decomposition for the expression. $$ \frac{3 x+11}{x^{2}-2 x-3} $$

Identify \(u\) and \(d v\) for finding the integral using integration by parts. (Do not evaluate the integral. $$ \int x^{2} e^{3 x} d x $$

Decide whether the integral is improper. Explain your reasoning. $$ \int_{0}^{1} \frac{2 x-5}{x^{2}-5 x+6} d x $$

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

Use the indicated formula from the table of integrals in this section to find the indefinite integral. $$ \int \frac{x}{\sqrt{2+3 x}} d x, \text { Formula } 19 $$

Write the partial fraction decomposition for the expression. $$ \frac{8 x+3}{x^{2}-3 x} $$

Identify \(u\) and \(d v\) for finding the integral using integration by parts. (Do not evaluate the integral. $$ \int x \ln 2 x d x $$

Decide whether the integral is improper. Explain your reasoning. $$ \int_{1}^{\infty} x^{2} d x $$

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

Use the indicated formula from the table of integrals in this section to find the indefinite integral. $$ \int \frac{4}{x^{2}-9} d x, \text { Formula } 29 $$

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

Identify \(u\) and \(d v\) for finding the integral using integration by parts. (Do not evaluate the integral. $$ \int \ln 4 x d x $$

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

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. $$ \int_{0}^{2} x^{3} d x, n=8 $$

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

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

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. $$ \int_{3}^{4} \frac{1}{\sqrt{x-3}} d x $$

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

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

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

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. $$ \int_{1}^{3}\left(4-x^{2}\right) d x, n=4 $$

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

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

Integration by parts to find the indefinite integral. $$ \int x^{2} e^{-x} d x $$

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. $$ \int_{1}^{2} \frac{1}{x} d x, n=8 $$

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

Write the partial fraction decomposition for the expression. $$ \frac{3 x^{2}-x+1}{x(x+1)^{2}} $$

Integration by parts to find the indefinite integral. $$ \int x^{2} e^{2 x} d x $$

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

Explain why the integral is improper and determine whether it diverges or converges. Evaluate the integral if it converses. $$ \int_{0}^{\infty} e^{-x} d x $$

Use the table of integrals in this section to find the indefinite integral. $$ \int \frac{1}{x(1+x)} d x $$

Write the partial fraction decomposition for the expression. $$ \frac{x+1}{3(x-2)^{2}} $$

Integration by parts to find the indefinite integral. $$ \int \ln 2 x d x $$

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. $$ \int_{0}^{4} \sqrt{x} d x, n=8 $$

Explain why the integral is improper and determine whether it diverges or converges. Evaluate the integral if it converses. $$ \int_{-\infty}^{0} e^{2 x} d x $$

Write the partial fraction decomposition for the expression. $$ \frac{3 x-4}{(x-5)^{2}} $$

Integration by parts to find the indefinite integral. $$ \int \ln x^{2} d x $$

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

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

Write the partial fraction decomposition for the expression. $$ \frac{8 x^{2}+15 x+9}{(x+1)^{3}} $$

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

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. $$ \int_{4}^{9} \sqrt{x} d x, n=8 $$