91Ó°ÊÓ

In each of Exercises \(41-54,\) determine whether the given improper integral is convergent or divergent. If it converges, then evaluate it. \(\int_{0}^{1} \frac{\arcsin (x)}{\sqrt{1-x^{2}}} d x\)

In each of Exercises \(53-56,\) make a substitution before applying the method of partial fractions to calculate the given integral. $$ \int \frac{\exp (x)}{\exp (2 x)-1} d x $$

Evaluate the given integral by converting the integrand to an expression in sines and cosines. $$ \int \cot ^{3}(x / 3) \csc (x / 3) d x $$

In each of Exercises \(41-54,\) determine whether the given improper integral is convergent or divergent. If it converges, then evaluate it. \(\int_{0}^{1} \frac{1}{\sqrt{\sqrt{x+1}-1} \sqrt{x+1}} d x\)

Make a substitution before applying the method of partial fractions to calculate the given integral. $$ \int \frac{2^{x+2}}{4^{x}-4} d x $$

Evaluate the given integral by converting the integrand to an expression in sines and cosines. $$ \int 2 \sin ^{2}(x) \tan (x) d x $$

Determine whether the given improper integral converges or diverges. If it converges, then evaluate it. $$ \int_{0}^{\infty} \frac{1}{e^{x}+1} d x $$

Evaluate each of the integrals. $$ \int\left(\frac{\ln (x)}{x}+\frac{\ln (x)}{x^{2}}\right) d x $$

Calculate the given integral. $$ \int \frac{2 x^{2}}{\sqrt{x^{2}-1}} d x $$

Evaluate \(\int_{-2}^{1} x(x+3)^{-1 / 2} d x .\) For the first step, integrate by parts with \(u=x\)