91Ó°ÊÓ

Solve the given problems by integration. Perform the integration \(\int \frac{x d x}{x^{2}+1}\) (a) by using the logarithmic formula, and (b) by trigonometric substitution. Compare results.

Integrate each of the given functions. $$\int \frac{4 e^{x} d x}{\left(1-e^{x}\right)^{2}}$$

Integrate each of the given functions. $$\int \frac{3 v^{2}-2 v}{v^{2}} d v$$

Solve the given problems by integration. Find the volume generated by revolving the first-quadrant region bounded by \(y=4 /\left(x^{4}+6 x^{2}+5\right)\) and \(x=2\) about the \(y\) -axis.

Solve the given problems by integration. To integrate \(\int x \ln (x+1) d x,\) the substitution \(t=x+1, d t=d x\) leads to an integral that can be done readily by parts. Perform this integration in this way.

Solve the given problems by integration. Find the area of a quarter circle of radius 2 by integrating \(\int_{0}^{2} \sqrt{4-x^{2}} d x\)

Solve the given problems by integration. $$\begin{aligned}&\text { Integrate: } \int \frac{\cos \theta}{\sin ^{2} \theta+2 \sin \theta-3} d \theta\\\&\text { substitution }u=\sin \theta .)\end{aligned}$$

Rewrite the given integrals so that they fit the form \(\int u^{n} d u,\) and identify \(u, n,\) and \(d u\). $$\int \sec ^{5} x \sin x d x$$

Solve the given problems by integration. Although \(\int \frac{d x}{1+\sin x}\) does not appear to fit a form for integration, show that it can be integrated by multiplying the numerator and the denominator by \(1-\sin x\)

Integrate each of the given functions. $$\int_{0}^{\pi / 12} \frac{\sec ^{2} 3 x}{4+\tan 3 x} d x$$