91Ó°ÊÓ

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

(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). $$ \int \frac{\ln x}{x \sqrt{4 \ln x-1}} d x $$

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

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

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

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

Evaluate the integral by making a substitution that converts the integrand to a rational function. \(\int \frac{e^{t}}{e^{2 t}-4} d t\)

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

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