91影视

Compute the definite integrals. Use a graphing utility to confirm your answers. $$ \int_{-\pi}^{\pi} x \sin x d x \text { (Express the answer in exact form.) } $$

Evaluate the integrals. If the integral diverges, answer "diverges."\(\int_{-1}^{1} \frac{d x}{\sqrt{1-x^{2}}}\)

Evaluate the integral \(\int \frac{d x}{x \sqrt{x^{2}-1}}\) using the substitution \(x=\sec \theta\). Next, evaluate the same integral using the substitution \(x=\csc \theta .\) Show that the results are equivalent.

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

Use tables to perform the integration. $$ \int \frac{3 x}{2 x+7} d x $$

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.) \(\int \tan x \sec ^{3} x d x\)

Use the method of partial fractions to evaluate the following integrals. \(\int \frac{3 x+4}{x^{3}-2 x-4} d x\) (Hint: Use the rational root theorem.)

Compute the definite integrals. Use a graphing utility to confirm your answers. $$ \int_{0}^{3} \ln \left(x^{2}+1\right) d x \text { (Express the answer in exact form.) } $$

Evaluate the integrals. If the integral diverges, answer "diverges."\(\int_{0}^{1} \frac{\ln x}{x} d x\)

Evaluate the integral \(\int \frac{x}{x^{2}+1} d x\) using the form \(\int \frac{1}{u} d u .\) Next, evaluate the same integral using \(x=\tan \theta .\) Are the results the same?