91Ó°ÊÓ

$$\text {Evaluate the following integrals.}$$ $$\int_{-\pi / 3}^{\pi / 3} \sqrt{\sec ^{2} \theta-1} d \theta$$

Evaluate the following definite integrals. $$\int_{0}^{1 / 3} \frac{d x}{\left(9 x^{2}+1\right)^{3 / 2}}$$

Find the volume of the described solid of revolution or state that it does not exist. The region bounded by \(f(x)=(x+1)^{-3 / 2}\) and the \(y\) -axis on the interval (-1,1] is revolved about the line \(x=-1.\)

Use the approaches discussed in this section to evaluate the following integrals. $$\int_{0}^{2} \frac{2}{x^{3}+3 x^{2}+3 x+1} d x$$

a. Evaluate \(\int x \ln x^{2} d x\) using the substitution \(u=x^{2}\) and evaluating \(\int \ln u d u\). b. Evaluate \(\int x \ln x^{2} d x\) using integration by parts. c. Verify that your answers to parts (a) and (b) are consistent.

$$\text {Evaluate the following integrals.}$$ $$\int_{-\pi / 4}^{\pi / 4} \tan ^{3} x \sec ^{2} x d x$$

Evaluate the following definite integrals. $$\int_{10 / \sqrt{3}}^{10} \frac{d x}{\sqrt{x^{2}-25}}$$

Use a computer algebra system to evaluate the following indefinite integrals. Assume that a is a positive real number. $$\int\left(x^{2}+a^{2}\right)^{-5 / 2} d x$$

a. Evaluate \(\int \tan x \sec ^{2} x d x\) using the substitution \(u=\tan x\) b. Evaluate \(\int \tan x \sec ^{2} x d x\) using the substitution \(u=\sec x\) c. Reconcile the results in parts (a) and (b).

Evaluate the following definite integrals. $$\int_{4 / \sqrt{3}}^{4} \frac{d x}{x^{2}\left(x^{2}-4\right)}$$