91Ó°ÊÓ

Evaluate the integrals in Exercises \(1-24\) using integration by parts. $$ \int x^{3} e^{x} d x $$

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. $$ \int_{0}^{\pi / 4} \frac{1+\sin \theta}{\cos ^{2} \theta} d \theta $$

Verify that the functions in Exercises \(11-16\) are probability density functions for a continuous random variable \(X\) over the given interval. Determine the specified probability. $$ f(x)=\left\\{\begin{array}{ll}{\frac{2}{x^{3}}} & {x>1} \\ {0} & {x \leq 1}\end{array}\right. \text { over }(-\infty, \infty), P(4 \leq X<9) $$

Use the table of integrals at the back of the book to evaluate the integrals. \(\int e^{2 t} \cos 3 t d t\)

Evaluate the integrals in Exercises \(1-22\) $$ \int_{0}^{\pi / 2} \sin ^{7} y d y $$

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

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

The integrals converge. Evaluate the integrals without using tables. $$\int_{0}^{2} \frac{s+1}{\sqrt{4-s^{2}}} d s$$

Verify that the functions in Exercises \(11-16\) are probability density functions for a continuous random variable \(X\) over the given interval. Determine the specified probability. $$ f(x)=\sin x \text { over }[0, \pi / 2], P\left(\frac{\pi}{6} < X \leq \frac{\pi}{4}\right) $$

Evaluate the integrals in Exercises \(1-22\) $$ \int 7 \cos ^{7} t d t $$