91Ó°ÊÓ

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

Compute the mean and median for a random variable with the probability density functions. $$f(x)=\left\\{\begin{array}{ll} \frac{2}{x^{3}} & x \geq 1 \\ 0 & x<1 \end{array}\right.$$

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. $$\int \frac{2 d x}{x \sqrt{1-4 \ln ^{2} x}}$$

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

Evaluate the integrals by using a substitution prior to integration by parts. $$\int \ln \left(x+x^{2}\right) d x$$

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

Use a substitution to change the integral into one you can find in the table. Then evaluate the integral. $$\int \frac{x^{2}+6 x}{\left(x^{2}+3\right)^{2}} d x$$

Evaluate the integrals without using tables. $$\int_{0}^{1} \frac{4 r d r}{\sqrt{1-r^{4}}}$$

Express the integrand as a sum of partial fractions and evaluate the integrals. $$\int \frac{1}{x^{4}+x} d x$$

Evaluate the integrals. $$\int_{0}^{\pi / 6} \sqrt{1+\sin x} d x$$ $$(\text {Hint: Multiply by } \sqrt{\frac{1-\sin x}{1-\sin x}})$$