91Ó°ÊÓ

Evaluate the integrals in Exercises \(1-24\) using integration by parts. $$ \int x^{5} e^{x} d x $$

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

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

Evaluate the integrals in Exercises \(1-22\) $$ \int 16 \sin ^{2} x \cos ^{2} x d x $$

Use the table of integrals at the back of the book to evaluate the integrals. \(\int x^{2} \tan ^{-1} x d x\)

In Exercises \(17-20\) , 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. $$ f(x)=4 e^{-2 x} \text { over }[0, c] $$

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

In Exercises \(17-20\) , 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. $$ f(x)=c x \sqrt{25-x^{2}} \text { over }[0,5] $$

Evaluate the integrals in Exercises \(1-24\) using integration by parts. $$ \int t^{2} e^{4 t} d t $$

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