91Ó°ÊÓ

Evaluate the following integrals. $$\int \frac{\sqrt{9-x^{2}}}{x^{2}} d x$$

Evaluate the following integrals or state that they diverge. $$\int_{-3}^{1} \frac{1}{(2 x+6)^{2 / 3}} d x$$

Evaluate the following definite integrals. $$\int_{2 / \sqrt{3}}^{2} z \sec ^{-1} z d z$$

Evaluate the following integrals. $$\int \frac{1-x}{1-\sqrt{x}} d x$$

Give the appropriate form of the partial fraction decomposition for the following functions. $$\frac{2}{x\left(x^{2}-6 x+9\right)}$$

Give the appropriate form of the partial fraction decomposition for the following functions. $$\frac{20 x}{(x-1)^{2}\left(x^{2}+1\right)}$$

Find the volume of the solid that is generated when the given region is revolved as described. The region bounded by \(f(x)=e^{-x}, x=\ln 2,\) and the coordinate axes is revolved about the \(y\) -axis.

Evaluate the following integrals. $$\int \frac{d x}{\sec x-1}$$

Evaluate the following integrals or state that they diverge. $$\int_{0}^{\pi / 2} \sec x \tan x d x$$

Determine whether the following statements are true and give an explanation or counterexample. a. The Trapezoid Rule is exact when used to approximate the definite integral of a linear function. b. If the number of subintervals used in the Midpoint Rule is increased by a factor of \(3,\) the error is expected to decrease by a factor of 8. c. If the number of subintervals used in the Trapezoid Rule is increased by a factor of \(4,\) the error is expected to decrease by a factor of 16