91Ó°ÊÓ

Using integration by parts. $$\int 4 x \sec ^{2} 2 x d x$$

Use the table of integrals at the back of the book to evaluate the integrals. $$\int \frac{\sqrt{x^{2}-4}}{x} d x$$

In Exercises \(9-16,\) express the integrand as a sum of partial fractions and evaluate the integrals. $$\int_{1 / 2}^{1} \frac{y+4}{y^{2}+y} d y$$

Evaluate the integrals. $$\int \frac{2 d x}{x^{3} \sqrt{x^{2}-1}}, x>1$$

Estimate the minimum number of sub intervals 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. ) $$\int_{-2}^{0}\left(x^{2}-1\right) d x$$

Converge. Evaluate the integrals without using tables. $$\int_{0}^{1} \frac{\theta+1}{\sqrt{\theta^{2}+2 \theta}} d \theta$$

Evaluate the integrals. $$\int_{0}^{\pi / 2} \sin ^{7} y d y$$

Use any method to evaluate the integrals. Most will require trigonometric substitutions, but some can be evaluated by other methods. $$\int \frac{x}{\sqrt{9-x^{2}}} d x$$

Estimate the minimum number of sub intervals 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. ) $$\int_{0}^{2}\left(t^{3}+t\right) d t$$

In Exercises \(9-16,\) express the integrand as a sum of partial fractions and evaluate the integrals. $$\int \frac{d t}{t^{3}+t^{2}-2 t}$$