91Ó°ÊÓ

Use the table of integrals at the back of the text to evaluate the integrals. $$\int \frac{\tan ^{-1} x}{x^{2}} d x$$

Evaluate the integrals $$\int_{0}^{\pi} 8 \sin ^{4} y \cos ^{2} y d y$$

Evaluate the integrals without using tables. $$\int_{-\infty}^{0} \theta e^{\theta} d \theta$$

In Exercises \(11-22,\) estimate the minimum number of subintervals needed to approximate the integrals with an error of magnitude less than \(10^{-4}\) by (a) the Trapezoidal Rule and (b) Simpson's Rule. (The integrals in Exercises \(11-18\) are the integrals from Exercises \(1-8 .\) ) $$\int_{0}^{2} \sin (x+1) d x$$

Evaluate the integrals $$\int 8 \cos ^{3} 2 \theta \sin 2 \theta d \theta$$

Express the integrand as a sum of partial fractions and evaluate the integrals. $$\int_{0}^{1} \frac{d x}{(x+1)\left(x^{2}+1\right)}$$

Evaluate the integrals using integration by parts. $$\int e^{\theta} \sin \theta d \theta$$

Use any method to evaluate the integrals in Exercises \(15-38 .\) Most will require trigonometric substitutions, but some can be evaluated by other methods. $$\int \frac{x^{3} d x}{\sqrt{x^{2}+4}}$$

Express the integrand as a sum of partial fractions and evaluate the integrals. $$\int_{1}^{\sqrt{3}} \frac{3 t^{2}+t+4}{t^{3}+t} d t$$

Use any method to evaluate the integrals in Exercises \(15-38 .\) Most will require trigonometric substitutions, but some can be evaluated by other methods. $$\int \frac{d x}{x^{2} \sqrt{x^{2}+1}}$$