Chapter 7

In Problems 1-36, use integration by parts to evaluate each integral. $$ \int x \sinh x d x $$

Use a CAS to evaluate the definite integrals in Problems \(31-40\). If the CAS does not give an exact answer in terms of elementary functions, then give a numerical approximation. $$ \int_{0}^{1} \operatorname{sech} \sqrt[3]{x} d x $$

In Problems 1-54, 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.

In Problems 1–40, use the method of partial fraction decomposition to perform the required integration. $$ \int \frac{x^{3}-8 x^{2}-1}{(x+3)\left(x^{2}-4 x+5\right)} d x $$

In Problems 1-36, use integration by parts to evaluate each integral. $$ \int \frac{\ln x}{\sqrt{x}} d x $$

Use a CAS to evaluate the definite integrals in Problems \(31-40\). If the CAS does not give an exact answer in terms of elementary functions, then give a numerical approximation. $$ \int_{0}^{\pi / 2} \sin ^{12} x d x $$

In Problems 1-54, perform the indicated integrations. \(\int \frac{t^{2} \cos \left(t^{3}-2\right)}{\sin ^{2}\left(t^{3}-2\right)} d t\)

In Problems 1–40, use the method of partial fraction decomposition to perform the required integration. $$ \int \frac{\left(\sin ^{3} t-8 \sin ^{2} t-1\right) \cos t}{(\sin t+3)\left(\sin ^{2} t-4 \sin t+5\right)} d t $$

In Problems 1-36, use integration by parts to evaluate each integral. $$ \int x \sinh x d x $$

Use a CAS to evaluate the definite integrals in Problems \(31-40\). If the CAS does not give an exact answer in terms of elementary functions, then give a numerical approximation. $$ \int_{0}^{\pi} \cos ^{4} \frac{x}{2} d x $$

In Problems 1-54, perform the indicated integrations. \(\int \frac{1+\cos 2 x}{\sin ^{2} 2 x} d x\)

In Problems 1–40, use the method of partial fraction decomposition to perform the required integration. $$ \int \frac{\cos t}{\sin ^{4} t-16} d t $$

In Problems 1-36, use integration by parts to evaluate each integral. $$ \int_{0}^{1} t(t-1)^{12} d t $$

Use a CAS to evaluate the definite integrals in Problems \(31-40\). If the CAS does not give an exact answer in terms of elementary functions, then give a numerical approximation. $$ \int_{1}^{4} \frac{\sqrt{t}}{1+t^{8}} d t $$

In Problems 1-54, perform the indicated integrations. \(\int \frac{t^{2} \cos ^{2}\left(t^{3}-2\right)}{\sin ^{2}\left(t^{3}-2\right)} d t\)

In Problems 1–40, use the method of partial fraction decomposition to perform the required integration. $$ \int \frac{x^{3}-4 x}{\left(x^{2}+1\right)^{2}} d x $$

In Problems 1-36, use integration by parts to evaluate each integral. $$ \int x 2^{x} d x $$

Use a CAS to evaluate the definite integrals in Problems \(31-40\). If the CAS does not give an exact answer in terms of elementary functions, then give a numerical approximation. $$ \int_{0}^{3} x^{4} e^{-x / 2} d x $$

In Problems 1-54, perform the indicated integrations. \(\int \frac{\csc ^{2} 2 t}{\sqrt{1+\cot 2 t}} d t\)

In Problems 1–40, use the method of partial fraction decomposition to perform the required integration. $$ \int \frac{(\sin t)\left(4 \cos ^{2} t-1\right)}{(\cos t)\left(1+2 \cos ^{2} t+\cos ^{4} t\right)} d t $$

In Problems 1-36, use integration by parts to evaluate each integral. $$ \int z a^{z} d z $$

Use a CAS to evaluate the definite integrals in Problems \(31-40\). If the CAS does not give an exact answer in terms of elementary functions, then give a numerical approximation. $$ \int_{0}^{\pi / 2} \frac{1}{1+2 \cos ^{5} x} d x $$

In Problems 1-54, perform the indicated integrations. \(\int \frac{e^{\tan ^{-1} 2 t}}{1+4 t^{2}} d t\)

In Problems 1–40, use the method of partial fraction decomposition to perform the required integration. $$ \int \frac{2 x^{3}+5 x^{2}+16 x}{x^{5}+8 x^{3}+16 x} d x $$

Use a CAS to evaluate the definite integrals in Problems \(31-40\). If the CAS does not give an exact answer in terms of elementary functions, then give a numerical approximation. $$ \int_{-\pi / 4}^{\pi / 4} \frac{x^{3}}{4+\tan x} d x $$

In Problems 1-54, perform the indicated integrations. \(\int(t+1) e^{-t^{2}-2 t-5} d t\)

In Problems 1–40, use the method of partial fraction decomposition to perform the required integration. $$ \int_{4}^{6} \frac{x-17}{x^{2}+x-12} d x $$

In Problems 37-48, apply integration by parts twice to evaluate each integral (see Examples 5 and 6). $$ \int x^{5} e^{x^{2}} d x $$

Use a CAS to evaluate the definite integrals in Problems \(31-40\). If the CAS does not give an exact answer in terms of elementary functions, then give a numerical approximation. $$ \int_{2}^{3} \frac{x^{2}+2 x-1}{x^{2}-2 x+1} d x $$

In Problems 1-54, perform the indicated integrations. \(\int \frac{y}{\sqrt{16-9 y^{4}}} d y \quad\)

In Problems 1–40, use the method of partial fraction decomposition to perform the required integration. $$ \int_{0}^{\pi / 4} \frac{\cos \theta}{\left(1-\sin ^{2} \theta\right)\left(\sin ^{2} \theta+1\right)^{2}} d \theta $$

In Problems 37-48, apply integration by parts twice to evaluate each integral (see Examples 5 and 6). $$ \int \ln ^{2} z d z $$

Use a CAS to evaluate the definite integrals in Problems \(31-40\). If the CAS does not give an exact answer in terms of elementary functions, then give a numerical approximation. $$ \int_{1}^{3} \frac{d u}{u \sqrt{2 u-1}} $$

In Problems 1-54, perform the indicated integrations. \(\int \cosh 3 x d x\)

In Problems 1–40, use the method of partial fraction decomposition to perform the required integration. $$ \int_{1}^{5} \frac{3 x+13}{x^{2}+4 x+3} d x $$

In Problems 41-48, the density of a rod is given. Find c so that the mass from 0 to \(c\) is equal to 1 . Whenever possible find an exact solution. If this is not possible, find an approximation for c. (See Examples 4 and 5 ). $$ \delta(x)=\frac{1}{x+1} $$

In Problems 1-54, perform the indicated integrations. \(\int x^{2} \sinh x^{3} d x\)

In Problems 41-44, solve the logistic differential equation representing population growth with the given initial condition. Then use the solution to predict the population size at time \(t=3\). $$ y^{\prime}=y(1-y), y(0)=0.5 $$

In Problems 1-54, perform the indicated integrations. \(\int \frac{5}{\sqrt{9-4 x^{2}}} d x\)

In Problems 41-44, solve the logistic differential equation representing population growth with the given initial condition. Then use the solution to predict the population size at time \(t=3\). $$ y^{\prime}=\frac{1}{10} y(12-y), y(0)=2 $$

In Problems 41-48, the density of a rod is given. Find c so that the mass from 0 to \(c\) is equal to 1 . Whenever possible find an exact solution. If this is not possible, find an approximation for c. (See Examples 4 and 5 ). $$ \delta(x)=\ln (x+1) $$

In Problems 41-44, solve the logistic differential equation representing population growth with the given initial condition. Then use the solution to predict the population size at time \(t=3\). $$ y^{\prime}=0.0003 y(8000-y), y(0)=1000 $$

In Problems 37-48, apply integration by parts twice to evaluate each integral (see Examples 5 and 6). $$ \int x^{2} \cos x d x $$

In Problems 1-54, perform the indicated integrations. \(\int \frac{d t}{2 t \sqrt{4 t^{2}-1}}\)

In Problems 41-44, solve the logistic differential equation representing population growth with the given initial condition. Then use the solution to predict the population size at time \(t=3\). $$ y^{\prime}=0.001 y(4000-y), y(0)=100 $$

In Problems 37-48, apply integration by parts twice to evaluate each integral (see Examples 5 and 6). $$ \int r^{2} \sin r d r $$

In Problems 1-54, perform the indicated integrations. \(\int_{0}^{\pi / 2} \frac{\sin x}{16+\cos ^{2} x} d x\)

Solve the logistic differential equation for an arbitrary constant of proportionality \(k\), capacity \(L\), and initial condition \(y(0)=y_{0}\).

In Problems 41-48, the density of a rod is given. Find c so that the mass from 0 to \(c\) is equal to 1 . Whenever possible find an exact solution. If this is not possible, find an approximation for c. (See Examples 4 and 5 ). $$ \delta(x)=\ln \left(x^{3}+1\right) $$