91Ó°ÊÓ

Use any method to evaluate the integrals in Exercises \(15-34 .\) Most will require trigonometric substitutions, but some can be evaluated by other methods. $$ \int \frac{x^{2} d x}{\left(x^{2}-1\right)^{5 / 2}}, \quad x>1 $$

The integrals in Exercises \(1-34\) converge. Evaluate the integrals without using tables. $$\int_{0}^{1}(-\ln x) d x$$

Use the table of integrals at the back of the book to evaluate the integrals in Exercises \(1-26 .\) $$ \int \cos \frac{\theta}{2} \cos 7 \theta d \theta $$

The integrals in Exercises \(1-44\) are in no particular order. Evaluate each integral using any algebraic method or trigonometric identity you think is appropriate. When necessary, use a substitution to reduce it to a standard form. $$ \int \frac{6 d y}{\sqrt{y}(1+y)} $$

Compute the mean and median for a random variable with the probability density functions. \(f(x)=\frac{1}{9} x^{2}\) over \([0,3]\)

In Exercises \(21-32,\) express the integrand as a sum of partial fractions and evaluate the integrals. $$ \int \frac{s^{4}+81}{s\left(s^{2}+9\right)^{2}} d s $$

Evaluate the integrals by using a substitution prior to integration by parts. $$ \int_{0}^{1} x \sqrt{1-x} d x $$

Evaluate the integrals. \(\int_{\pi / 3}^{\pi / 2} \frac{\sin ^{2} x}{\sqrt{1-\cos x}} d x\)

Evaluate the integrals by using a substitution prior to integration by parts. $$ \int_{0}^{\pi / 3} x \tan ^{2} x d x $$

Use any method to evaluate the integrals in Exercises \(15-34 .\) Most will require trigonometric substitutions, but some can be evaluated by other methods. $$ \int \frac{\left(1-x^{2}\right)^{3 / 2}}{x^{6}} d x $$