91Ó°ÊÓ

Use any method to evaluate the integrals. Most will require trigonometric substitutions, but some can be evaluated by other methods. \(\int \frac{8 d w}{w^{2} \sqrt{4-w^{2}}}\)

Evaluate the integrals using integration by parts. $$\int x^{5} e^{x} d x$$

Evaluate the integrals without using tables. $$\int_{0}^{\infty} \frac{d v}{\left(1+v^{2}\right)\left(1+\tan ^{-1} v\right)}$$

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

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. $$\int \frac{d \theta}{\sec \theta+\tan \theta}$$

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 \(11-18 \text { are the integrals from Exercises } 1-8 .)\) $$\int_{0}^{3} \sqrt{x+1} d x$$

Evaluate the integrals. $$\int 16 \sin ^{2} x \cos ^{2} x d x$$

Find the value of the constant \(c\) so that the given function is a probability density function for a random variable over the specified interval. $$f(x)=c x \sqrt{25-x^{2}} \text { over }[0,5]$$

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

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 \(11-18 \text { are the integrals from Exercises } 1-8 .)\) $$\int_{0}^{3} \frac{1}{\sqrt{x+1}} d x$$