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Compute the following integrals using the guidelines for integrating powers of trigonometric functions. Use a CAS to check the solutions. (Note: Some of the problems may be done using techniques of integration learned previously.) \(\int \sin ^{5} x \cos ^{2} x d x\)

Find the integral by using the simplest method. Not all problems require integration by parts. $$ \int x \cosh x d x $$

Determine the convergence of each of the following integrals by comparison with the given integral. If the integral converges, find the number to which it converges.\(\int_{1}^{\infty} \frac{d x}{\sqrt{x}+1} ;\) compare with \(\int_{1}^{\infty} \frac{d x}{2 \sqrt{x}}\)

Evaluate \(\int_{0}^{1} \frac{d x}{1+x^{2}}\) exactly and show that the result is \(\pi / 4\). Then, find the approximate value of the integral using the trapezoidal rule with \(n=4\) subdivisions. Use the result to approximate the value of \(\pi\).

Use the method of partial fractions to evaluate each of the following integrals. \(\int \frac{d x}{x^{4}-10 x^{2}+9}\)

Compute the definite integrals. Use a graphing utility to confirm your answers. $$ \int_{1 / e}^{1} \ln x d x $$

Approximate \(\int_{2}^{4} \frac{1}{\ln x} d x\) using the midpoint rule with four subdivisions to four decimal places.

Evaluate the integrals. If the integral diverges, answer "diverges."\(\int_{1}^{\infty} \frac{d x}{x^{e}}\)

Use tables to evaluate the integrals. You may need to complete the square or change variables to put the integral into a form given in the table. $$ \int \frac{e^{x}}{\sqrt{e^{2 x}-4}} d x $$

Compute the following integrals using the guidelines for integrating powers of trigonometric functions. Use a CAS to check the solutions. (Note: Some of the problems may be done using techniques of integration learned previously.) \(\int \sin ^{3} x \cos ^{3} x d x\)