Problem 80
Evaluate the following integrals. $$\int_{-2}^{-1} \sqrt{-4 x-x^{2}} d x$$
Problem 80
Skydiving A skydiver in free fall subject to gravitational acceleration and air resistance has a velocity given by \(v(t)=v_{T}\left(\frac{e^{a t}-1}{e^{a t}+1}\right),\) where \(v_{T}\) is the terminal velocity and \(a>0\) is a physical constant. Find the distance that the skydiver falls after \(t\) seconds, which is \(d(t)=\int_{0}^{t} v(y) d y\)
Problem 80
Preliminary steps The following integrals require a preliminary step such as a change of variables before using the method of partial fractions. Evaluate these integrals. $$\int \sqrt{e^{x}+1} d x \,(\text { Hint: Let } u=\sqrt{e^{x}+1}.)$$
Problem 80
Determine whether the following integrals converge or diverge. $$\int_{2}^{\infty} \frac{x^{3}}{x^{4}-x-1} d x$$
Problem 81
Determine whether the following integrals converge or diverge. $$ \int_{1}^{\infty} \frac{\sin ^{2} x}{x^{2}} d x $$
Problem 81
Evaluate the following integrals. $$\int \frac{x^{4}+2 x^{3}+5 x^{2}+2 x+1}{x^{5}+2 x^{3}+x} d x$$
Problem 81
Preliminary steps The following integrals require a preliminary step such as a change of variables before using the method of partial fractions. Evaluate these integrals. $$\int \frac{\left(e^{3 x}+e^{2 x}+e^{x}\right)}{\left(e^{2 x}+1\right)^{2}} d x$$
Problem 82
Evaluate the following integrals. $$\int \frac{d x}{1+\tan x}$$
Problem 82
The following integrals may require more than one table look-up. Evaluate the integrals using a table of integrals, and then check your answer with a computer algebra system. $$\int \frac{\sin ^{-1} a x}{x^{2}} d x, a>0$$
Problem 82
A family of exponentials The curves \(y=x e^{-a x}\) are shown in the figure for \(a\)=1,2, and 3. Figure cannot copy a. Find the area of the region bounded by \(y=x e^{-x}\) and the \(x\) -axis on the interval [0,4] b. Find the area of the region bounded by \(y=x e^{-a x}\) and the \(x\) -axis on the interval \([0,4],\) where \(a>0.\) c. Find the area of the region bounded by \(y=x e^{-a x}\) and the \(x\) -axis on the interval \([0, b] .\) Because this area depends on \(a\) and \(b,\) we call it \(A(a, b)\). d. Use part (c) to show that \(A(1, \ln b)=4 A\left(2, \frac{\ln b}{2}\right)\). e. Does this pattern continue? Is it true that \(A(1, \ln b)=a^{2} A(a,(\ln b) / a) ?\)
