Chapter 7

$$ \text { use integration by parts to evaluate each integral. } $$ $$ \int t^{5} \ln \left(t^{7}\right) d t $$

Use the method of completing the square, along with a trigonometric substitution if needed, to evaluate each integral. \(\int \frac{d x}{\sqrt{4 x-x^{2}}}\)

A tank of capacity 100 gallons is initially full of pure alcohol. The flow rate of the drain pipe is 5 gallons per minute; the flow rate of the filler pipe can be adjusted to \(c\) gallons per minute. An unlimited amount of \(25 \%\) alcohol solution can be brought in through the filler pipe. Our goal is to reduce the amount of alcohol in the tank so that it will contain 100 gallons of \(50 \%\) solution. Let \(T\) be the number of minutes required to accomplish the desired change. (a) Evaluate \(T\) if \(c=5\) and both pipes are opened. (b) Evaluate \(T\) if \(c=5\) and we first drain away a sufficient amount of the pure alcohol and then close the drain and open the filler pipe. (c) For what values of \(c\) (if any) would strategy (b) give a faster time than (a)? (d) Suppose that \(c=4\). Determine the equation for \(T\) if we initially open both pipes and then close the drain.

Use the method of partial fraction decomposition to perform the required integration. \(\int \frac{3 x+2}{x^{3}+3 x^{2}+3 x+1} d x\)

Perform the indicated integrations. $$ \int \tan ^{5}\left(\frac{\theta}{2}\right) d \theta $$

Perform the indicated integrations. $$ \int \frac{3 e^{2 x}}{\sqrt{1-e^{2 x}}} d x $$1

$$ \text { use integration by parts to evaluate each integral. } $$ $$ \int_{\pi / 6}^{\pi / 2} x \csc ^{2} x d x $$

Use the method of completing the square, along with a trigonometric substitution if needed, to evaluate each integral. \(\int \frac{x}{\sqrt{4 x-x^{2}}} d x\)

The differential equation for a falling body near the earth's surface with air resistance proportional to the velocity \(v\) is \(d v / d t=-g-a v\), where \(g=32\) feet per second per second is the acceleration of gravity and \(a>0\) is the drag coefficient. Show each of the following: (a) \(v(t)=\left(v_{0}-v_{\infty}\right) e^{-a t}+v_{\infty}\), where \(v_{0}=v(0)\), and $$ v_{\infty}=-g / a=\lim _{t \rightarrow \infty} v(t) $$ is the so-called terminal velocity. (b) If \(y(t)\) denotes the altitude, then $$ y(t)=y_{0}+t v_{\infty}+(1 / a)\left(v_{0}-v_{\infty}\right)\left(1-e^{-a t}\right) $$

Use the method of partial fraction decomposition to perform the required integration. \(\int \frac{x^{6}}{(x-2)^{2}(1-x)^{5}} d x\)

Perform the indicated integrations. $$ \int \frac{x^{3}}{x^{4}+4} d x $$

Perform the indicated integrations. $$ \int \cot ^{5} 2 t d t $$

$$ \text { use integration by parts to evaluate each integral. } $$ $$ \int_{\pi / 6}^{\pi / 4} x \sec ^{2} x d x $$

Use the method of completing the square, along with a trigonometric substitution if needed, to evaluate each integral. \(\int \frac{2 x+1}{x^{2}+2 x+2} d x\)

Use the method of partial fraction decomposition to perform the required integration. \(\int \frac{3 x^{2}-21 x+32}{x^{3}-8 x^{2}+16 x} d x\)

Perform the indicated integrations. $$ \int \tan ^{-3} x \sec ^{4} x d x $$

$$ \text { use integration by parts to evaluate each integral. } $$ $$ \int x^{5} \sqrt{x^{3}+4} d x $$

Use the method of completing the square, along with a trigonometric substitution if needed, to evaluate each integral. \(\int \frac{2 x-1}{x^{2}-6 x+18} d x\)

Use the method of partial fraction decomposition to perform the required integration. \(\int \frac{x^{2}+19 x+10}{2 x^{4}+5 x^{3}} d x\)

Use the table of integrals on the inside back cover, perhaps combined with a substitution, to evaluate the given integrals. $$ \int \frac{\operatorname{sech} \sqrt{x}}{\sqrt{x}} d x $$

Perform the indicated integrations. $$ \int_{0}^{\pi / 6} 2^{\cos x} \sin x d x $$

Perform the indicated integrations. $$ \int \tan ^{1 / 2} x \sec ^{4} x d x $$

$$ \text { use integration by parts to evaluate each integral. } $$ $$ \int x^{13} \sqrt{x^{7}+1} d x $$

The region bounded by \(y=1 /\left(x^{2}+2 x+5\right), y=0\), \(x=0\), and \(x=1\), is revolved about the \(x\) -axis. Find the volume of the resulting solid.

For the differential equation \(\frac{d y}{d x}-\frac{y}{x}=x^{2}, x>0\), the integrating factor is \(e^{\int(-1 / x) d x} .\) The general antiderivative \(\int\left(-\frac{1}{x}\right) d x\) is equal to \(-\ln x+C .\) (a) Multiply both sides of the differential equation by \(\exp \left(\int\left(-\frac{1}{x}\right) d x\right)=\exp (-\ln x+C), \quad\) and show that \(\exp (-\ln x+C)\) is an integrating factor for every value of \(C .\) (b) Solve the resulting equation for \(y\), and show that the solution agrees with the solution obtained when we assumed that \(C=0\) in the integrating factor.

Use the method of partial fraction decomposition to perform the required integration. \(\int \frac{2 x^{2}+x-8}{x^{3}+4 x} d x\)

Use the table of integrals on the inside back cover, perhaps combined with a substitution, to evaluate the given integrals. $$ \int \frac{\cos t \sin t}{\sqrt{2 \cos t+1}} d t $$

Perform the indicated integrations. $$ \int \frac{\sin x-\cos x}{\sin x} d x $$

Perform the indicated integrations. $$ \int \tan ^{3} x \sec ^{2} x d x $$

Multiply both sides of the equation \(\frac{d y}{d x}+P(x) y=Q(x)\) by the factor \(e^{\int P(x) d x+C}\). (a) Show that \(e^{\int P(x) d x+C}\) is an integrating factor for every value of \(C\). (b) Solve the resulting equation for \(y\), and show that it agrees with the general solution given before Example 1 .

Use the method of partial fraction decomposition to perform the required integration. \(\int \frac{3 x+2}{x(x+2)^{2}+16 x} d x\)

Perform the indicated integrations. $$ \int \frac{\sin (4 t-1)}{1-\sin ^{2}(4 t-1)} d t $$

Perform the indicated integrations. $$ \int \tan ^{3} x \sec ^{-1 / 2} x d x $$

$$ \text { use integration by parts to evaluate each integral. } $$ $$ \int x^{3} \sqrt{4-x^{2}} d x $$

Use the method of partial fraction decomposition to perform the required integration. \(\int \frac{2 x^{2}-3 x-36}{(2 x-1)\left(x^{2}+9\right)} d x\)

$$ \text { Find } \int_{-\pi}^{\pi} \cos m x \cos n x d x, m \neq n ; m, n \text { integers. } $$

$$ \text { use integration by parts to evaluate each integral. } $$ $$ \int \frac{z^{7}}{\left(4-z^{4}\right)^{2}} d z $$

Find \(\int_{0}^{3} \frac{x^{3} d x}{\sqrt{9+x^{2}}}\) by making the substitutions \(u=\sqrt{9+x^{2}}, \quad u^{2}=9+x^{2}, \quad 2 u d u=2 x d x\)

Perform the indicated integrations. $$ \int e^{x} \sec ^{2}\left(e^{x}\right) d x $$

$$ \text { Find } \int_{-L}^{L} \cos \frac{m \pi x}{L} \cos \frac{n \pi x}{L} d x, m \neq n, m, n \text { integers. } $$

$$ \text { use integration by parts to evaluate each integral. } $$ $$ \int x \cosh x d x $$

Find \(\int \frac{\sqrt{4-x^{2}}}{x} d x\) by (a) the substitution \(u=\sqrt{4-x^{2}}\) and (b) a trigonometric substitution. Then reconcile your answers.

Use the method of partial fraction decomposition to perform the required integration. \(\int \frac{1}{(x-1)^{2}(x+4)^{2}} d x\)

Use a CAS to evaluate the definite integrals. If the CAS does not give an exact answer in terms of elementary functions, then give a numerical approximation. $$ \int_{0}^{\pi} \frac{\cos ^{2} x}{1+\sin x} d x $$

Perform the indicated integrations. $$ \int \frac{\sec ^{3} x+e^{\sin x}}{\sec x} d x $$

$$ \text { use integration by parts to evaluate each integral. } $$ $$ \int x \sinh x d x $$

Use the method of partial fraction decomposition to perform the required integration. \(\int \frac{x^{3}-8 x^{2}-1}{(x+3)\left(x^{2}-4 x+5\right)} d x\)

Use a CAS to evaluate the definite integrals. If the CAS does not give an exact answer in terms of elementary functions, then give a numerical approximation. $$ \int_{0}^{1} \operatorname{sech} \sqrt[3]{x} d x $$

Perform the indicated integrations. $$ \int \frac{(6 t-1) \sin \sqrt{3 t^{2}-t-1}}{\sqrt{3 t^{2}-t-1}} d t $$

The region bounded by \(y=\sin ^{2}\left(x^{2}\right), y=0\), and \(x=\sqrt{\pi / 2}\) is revolved about the \(y\) -axis. Find the volume of the resulting solid.