91Ó°ÊÓ

Use a substitution to change the integral into one you can find in the table. Then evaluate the integral. $$\int \tan ^{-1} \sqrt{y} d y$$

Use an appropriate substitution and then a trigonometric substitution to evaluate the integrals. $$\int_{\ln (3 / 4)}^{\ln (4 / 3)} \frac{e^{t} d t}{\left(1+e^{2 t}\right)^{3 / 2}}$$

Some integrals do not require integration by parts. $$\int \frac{(\ln x)^{3}}{x} d x$$

Evaluate the integrals. $$\int \sec ^{3} x \tan ^{3} x d x$$

In Exercises \(33-38,\) perform long division on the integrand, write the proper fraction as a sum of partial fractions, and then evaluate the integral. $$\int \frac{16 x^{3}}{4 x^{2}-4 x+1} d x$$

Use integration, the Direct Comparison Test, or the Limit Comparison Test to test the integrals for convergence. If more than one method applies, use whatever method you prefer. $$\int_{0}^{1} \frac{\ln x}{x^{2}} d x$$

Use numerical integration to estimate the value of $$\sin ^{-1} 0.6=\int_{0}^{0.6} \frac{d x}{\sqrt{1-x^{2}}}$$ For reference, \(\sin ^{-1} 0.6=0.64350\) to five decimal places.

In Exercises \(33-38,\) perform long division on the integrand, write the proper fraction as a sum of partial fractions, and then evaluate the integral. $$\int \frac{y^{4}+y^{2}-1}{y^{3}+y} d y$$

Use a substitution to change the integral into one you can find in the table. Then evaluate the integral. \(\int \frac{1}{\sqrt{x^{2}+2 x+5}} d x\) (Hint: Complete the square.)

Use an appropriate substitution and then a trigonometric substitution to evaluate the integrals. $$\int_{1 / 12}^{1 / 4} \frac{2 d t}{\sqrt{t}+4 t \sqrt{t}}$$