91Ó°ÊÓ

Evaluate the integral. $$\int \tan x \sec ^{5} x d x$$

Evaluate the integral. $$\int_{0}^{2} \ln \left(x^{2}+1\right) d x$$

Use Simpson's rule approximation \(S_{10}\) to approximate the length of the curve over the stated interval. Express your answers to at least four decimal places. $$y=\sin x \text { from } x=0 \text { to } x=\pi$$

Evaluate the integral by making a substitution that converts the integrand to a rational function. $$\int \frac{\cos \theta}{\sin ^{2} \theta+4 \sin \theta-5} d \theta$$

Make the \(u\) -substitution and evaluate the resulting definite integral. $$\begin{aligned} &\int_{0}^{+\infty} \frac{e^{-x}}{\sqrt{1-e^{-x}}} d x ; u=1-e^{-x}\\\ &[\text {Note}: u \rightarrow 1 \text { as } x \rightarrow+\infty .] \end{aligned}$$

(a).Make an appropriate \(u\) -substitution, and then use the Endpaper Integral Table to evaluate the integral. (b) If you have a CAS, use it to evaluate the integral (no substitution), and then confirm that the result is equivalent to that in part (a). $$\int \frac{x}{16 x^{4}-1} d x$$

Determine whether the statement is true or false. Explain your answer. The main goal in integration by parts is to choose \(u\) and \(d v\) to obtain a new integral that is easier to evaluate than the original.

Evaluate the integral. $$\int \frac{d x}{\sqrt{3+2 x-x^{2}}}$$

Evaluate the integral. $$\int \sec ^{4} x d x$$

Make the \(u\) -substitution and evaluate the resulting definite integral. $$\int_{0}^{+\infty} \frac{e^{-x}}{\sqrt{1-e^{-2 x}}} d x ; u=e^{-x}$$