91Ó°ÊÓ

Use differentiation to match the antiderivative with the correct integral. [Integrals are labeled (a), (b), (c), and \((\mathbf{d}) .]\) (a) \(\int \sin x \tan ^{2} x d x\) (b) \(8 \int \cos ^{4} x d x\) (c) \(\int \sin x \sec ^{2} x d x\) (d) \(\int \tan ^{4} x d x\) $$ y=x-\tan x+\frac{1}{3} \tan ^{3} x $$

Write the form of the partial fraction decomposition of the rational expression. Do not solve for the constants. $$ \frac{2 x-3}{x^{3}+10 x} $$

In Exercises \(3-6,\) evaluate the limit (a) using techniques from Chapters 1 and 3 and (b) using L'Hôpital's Rule. \(\lim _{x \rightarrow 3} \frac{2(x-3)}{x^{2}-9}\)

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

In Exercises \(3-6,\) evaluate the limit (a) using techniques from Chapters 1 and 3 and (b) using L'Hôpital's Rule. \(\lim _{x \rightarrow 0} \frac{\sin 4 x}{2 x}\)

Write the form of the partial fraction decomposition of the rational expression. Do not solve for the constants. $$ \frac{2 x-1}{x\left(x^{2}+1\right)^{2}} $$

Find the indefinite integral using the substitution \(x=5 \sin \theta\) $$ \int \frac{1}{\left(25-x^{2}\right)^{3 / 2}} d x $$

In Exercises \(5-8,\) identify \(u\) and \(d v\) for finding the integral using integration by parts. (Do not evaluate the integral.) $$ \int x e^{2 x} d x $$

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

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