91Ó°ÊÓ

Use the table of integrals at the back of the book to evaluate the integrals. $$\int \frac{\sqrt{4-x^{2}}}{x} d x$$

Evaluate the integrals using integration by parts. $$\int x \sec ^{2} x d x$$

Verify that the functions are probability density functions for a continuous random variable \(X\) over the given interval. Determine the specified probability. $$f(x)=\frac{3}{2} x(2-x) \text { over }[0,1], P(0.5>X)$$

Use the table of integrals at the back of the book to evaluate the integrals. $$\int \frac{\sqrt{x^{2}-4}}{x} d x$$

Evaluate the integrals without using tables. $$\int_{-\infty}^{\infty} \frac{x d x}{\left(x^{2}+4\right)^{3 / 2}}$$

Estimate the minimum number of subintervals needed to approximate the integrals with an error of magnitude less than \(10^{-4}\) by (a) the Trapezoidal Rule and (b) Simpson's Rule. (The integrals in Exercises \(11-18 \text { are the integrals from Exercises } 1-8 .)\) $$\int_{-2}^{0}\left(x^{2}-1\right) d x$$

Evaluate the integrals. $$\int_{0}^{\pi / 2} \sin ^{2} x d x$$

Evaluate the integrals. \(\int \frac{2 d x}{x^{3} \sqrt{x^{2}-1}}, \quad x>1\)

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. $$\int \csc t \sin 3 t d t$$

Express the integrand as a sum of partial fractions and evaluate the integrals. $$\int_{1 / 2}^{1} \frac{y+4}{y^{2}+y} d y$$