91Ó°ÊÓ

Using the trigonometric substitution \(x=8 \sec \theta, x \geq 8\) and \(0 < \theta \leq \frac{\pi}{2},\) express tan \(\theta\) in terms of \(x\)

For what values of \(p\) does \(\int_{1}^{\infty} x^{-p} d x\) converge?

How would you evaluate \(\int \cos ^{2} x \sin ^{3} x d x ?\)

How would you choose \(d v\) when evaluating \(\int x^{n} e^{a x} d x\) using integration by parts?

Set up the appropriate form of the partial fraction decomposition for the following expressions. Do not find the values of the unknown constants. $$\frac{4 x+1}{4 x^{2}-1}$$

Evaluate \(\int \frac{2 x+1}{x^{2}+1} d x\) using the following steps. a. Fill in the blanks: By splitting the integrand into two fractions, we have \(\int \frac{2 x+1}{x^{2}+1} d x=\int \quad d x+\int \quad d x\) b. Evaluate the two integrals on the right side of the equation in part (a).

Choosing an integration strategy Identify a technique of integration for evaluating the following integrals. If necessary, explain how to first simplify the integrals before applying the suggested technique of integration. You do not need to evaluate the integrals. $$\int \frac{\cos ^{5} x \sin ^{4} x}{1-\sin ^{2} x} d x$$

Evaluate the following integrals or state that they diverge. $$\int_{3}^{\infty} \frac{d x}{x^{2}}$$

Trigonometric substitutions Evaluate the following integrals using trigonometric substitution. \(\int_{0}^{5 / 2} \frac{d x}{\sqrt{25-x^{2}}} \quad(\text {Hint}:\) Check your answer without using trigonometric substitution.)

Set up the appropriate form of the partial fraction decomposition for the following expressions. Do not find the values of the unknown constants. $$\frac{x+3}{(x-5)^{2}}$$