91Ó°ÊÓ

In Exercises 9-24, find the integral. (Note: Solve by the simplest method-not all require integration by parts.) $$ \int x e^{-2 x} d x $$

Use Wallis's Formulas to evaluate the integral. $$ \int_{0}^{\pi / 2} \cos ^{3} x d x $$

Use Wallis's Formulas to evaluate the integral. $$ \int_{0}^{\pi / 2} \sin ^{6} x d x $$

Use Wallis's Formulas to evaluate the integral. $$ \int_{0}^{\pi / 2} \sin ^{7} x d x $$

Find the integral. $$ \int \frac{1}{4+4 x^{2}+x^{4}} d x $$

In Exercises 27-30, verify the integration formula. $$ \int \frac{u^{2}}{(a+b u)^{2}} d u=\frac{1}{b^{3}}\left(b u-\frac{a^{2}}{a+b u}-2 a \ln |a+b u|\right)+C $$

In Exercises \(41-44,\) find or evaluate the integral using substitution first, then using integration by parts. $$ \int \sin \sqrt{x} d x $$

Given continuous functions \(f\) and \(g\) such that \(0 \leq f(x) \leq g(x)\) on the interval \([a, \infty),\) prove the following. (a) If \(\int_{a}^{\infty} g(x) d x\) converges, then \(\int_{a}^{\infty} f(x) d x\) converges. (b) If \(\int_{a}^{\infty} f(x) d x\) diverges, then \(\int_{a}^{\infty} g(x) d x\) diverges.

Fluid Force Find the fluid force on a circular observation window of radius 1 foot in a vertical wall of a large water-filled tank at a fish hatchery when the center of the window is (a) 3 feet and (b) \(d\) feet \((d>1)\) below the water's surface. Use trigonometric substitution to evaluate the one integral.

Sketch the graph of the hypocycloid of four cusps \(x^{2 / 3}+y^{2 / 3}=4\) and find its perimeter