91Ó°ÊÓ

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

In Exercises \(21-32,\) express the integrand as a sum of partial fractions and evaluate the integrals. $$\int \frac{y^{2}+2 y+1}{\left(y^{2}+1\right)^{2}} d y$$

The integrals converge. Evaluate the integrals without using tables. $$\int_{-\infty}^{0} e^{-|x|} d x$$

Use the table of integrals at the back of the book to evaluate the integrals. \(\int 8 \sin 4 t \sin \frac{t}{2} d t\)

Use any method to evaluate the integrals. Most will require trigonometric substitutions, but some can be evaluated by other methods. $$\int_{0}^{\sqrt{3} / 2} \frac{4 x^{2} d x}{\left(1-x^{2}\right)^{3 / 2}}$$

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. $$ \int_{0}^{\pi / 2} \sqrt{1-\cos \theta} d \theta $$

Evaluate the integrals in Exercises \(1-24\) using integration by parts. $$ \int e^{2 x} \cos 3 x d x $$

Show that if the exponentially decreasing function $$ f(x)=\left\\{\begin{array}{cl}{0} & {\text { if } x<0} \\ {A e^{-c x}} & {\text { if } x \geq 0}\end{array}\right. $$ is a probability density function, then \(A=c\)

Evaluate the integrals in Exercises \(23-32\) $$ \int_{0}^{2 \pi} \sqrt{\frac{1-\cos x}{2}} d x $$

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