91Ó°ÊÓ

Evaluate the integrals in Exercises \(33-50\) $$ \int \sec ^{3} x \tan ^{3} x d x $$

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. $$\int_{0}^{1} \frac{\ln x}{x^{2}} d x$$

Evaluate the integrals in Exercises \(31-52 .\) Some integrals do not require integration by parts. $$ \int x^{3} e^{x^{4}} d x $$

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. $$\int \frac{y^{4}+y^{2}-1}{y^{3}+y} d y$$

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.

Evaluate the integrals in Exercises \(33-50\) $$ \int \sec ^{2} x \tan ^{2} x d x $$

The integrals in Exercises \(1-40\) are in no particular order. Evaluate each integral using any algebraic method or trigonometric identity you think is appropriate, and then use a substitution to reduce it to a standard form. $$ \int \frac{2 \theta^{3}-7 \theta^{2}+7 \theta}{2 \theta-5} d \theta $$

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. $$\int_{1 / 12}^{1 / 4} \frac{2 d t}{\sqrt{t}+4 t \sqrt{t}}$$

Use numerical integration to estimate the value of $$ \pi=4 \int_{0}^{1} \frac{1}{1+x^{2}} d x $$