91Ó°ÊÓ

Find the area of the region between the \(x\) -axis and the curve \(y=8 /\left(x^{2}-4\right)\) for \(x \geq 4\).

(a) Use the hyperbolic substitution \(x=3\) sinh \(u\), the identity \(\cosh ^{2} u-\sinh ^{2} u=1,\) and Theorem 6.9 .4 to evaluate $$\int \frac{d x}{\sqrt{x^{2}+9}}$$ b) Evaluate the integral in part (a) using a trigonometric substitution and show that the result agrees with that obtained in part (a).

Suppose that the region between the \(x\) -axis and the curve \(y=e^{-x}\) for \(x \geq 0\) is revolved about the \(x\) -axis. (a) Find the volume of the solid that is generated. (b) Find the surface area of the solid.

Evaluate the integral. $$\int \cot ^{3} x d x$$

(a) Complete the square, make an appropriate \(u\) -substitution, and then use the End paper Integral Table to evaluate the integral. (b) If you have a CAS, use it to evaluate the integral (no substitution or square completion), and then confirm that the result is equivalent to that in part (a). $$\int \frac{x}{\sqrt{5+4 x-x^{2}}} d x$$

Suppose that \(a x^{2}+b x+c\) is a quadratic polynomial and that the integration $$\int \frac{1}{a x^{2}+b x+c} d x$$ produces a function with no inverse tangent terms. What does this tell you about the roots of the polynomial?

Evaluate the integral using tabular integration by parts. $$\int e^{a x} \sin b x d x$$

Evaluate the integral using tabular integration by parts. $$\int e^{-3 \theta} \sin 5 \theta d \theta$$

(a) Complete the square, make an appropriate \(u\) -substitution, and then use the End paper Integral Table to evaluate the integral. (b) If you have a CAS, use it to evaluate the integral (no substitution or square completion), and then confirm that the result is equivalent to that in part (a). $$\int \frac{x}{x^{2}+6 x+13} d x$$

Evaluate the integral. $$\int \csc ^{4} x d x$$