Chapter 7

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

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

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. $$ \int_{0}^{\pi / 2} \sin ^{12} x d x $$

Perform the indicated integrations. $$ \int \frac{t^{2} \cos \left(t^{3}-2\right)}{\sin ^{2}\left(t^{3}-2\right)} d t $$

$$ \text { use integration by parts to evaluate each integral. } $$ $$ \int x(3 x+10)^{49} d x $$

Use the method of partial fraction decomposition to perform the required integration. \(\int \frac{\cos t}{\sin ^{4} t-16} d t\)

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. $$ \int_{0}^{\pi} \cos ^{4} \frac{x}{2} d x $$

Perform the indicated integrations. $$ \int \frac{1+\cos 2 x}{\sin ^{2} 2 x} d x $$

Show that $$ \lim _{n \rightarrow \infty} \cos \frac{x}{2} \cos \frac{x}{4} \cos \frac{x}{8} \cdots \cos \frac{x}{2^{n}}=\frac{\sin x}{x} $$ by completing the following steps. $$ \begin{array}{l} \text { (a) } \cos \frac{x}{2} \cos \frac{x}{4} \cdots \cos \frac{x}{2^{n}}= \\\ {\left[\cos \frac{1}{2^{n}} x+\cos \frac{3}{2^{n}} x+\cdots+\cos \frac{2^{n}-1}{2^{n}} x\right] \frac{1}{2^{n-1}}} \end{array} $$ (See Problem 46 of Section 1.8.) (b) Recognize a Riemann sum leading to a definite integral. (c) Evaluate the definite integral.

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

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

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. $$ \int_{1}^{4} \frac{\sqrt{t}}{1+t^{8}} d t $$

Perform the indicated integrations. $$ \int \frac{t^{2} \cos ^{2}\left(t^{3}-2\right)}{\sin ^{2}\left(t^{3}-2\right)} d t $$

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

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

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. $$ \int_{0}^{3} x^{4} e^{-x / 2} d x $$

Perform the indicated integrations. $$ \int \frac{\csc ^{2} 2 t}{\sqrt{1+\cot 2 t}} d t $$

$$ \text { use integration by parts to evaluate each integral. } $$ $$ \int z a^{z} d z $$

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. If the CAS does not give an exact answer in terms of elementary functions, then give a numerical approximation. $$ \int_{0}^{\pi / 2} \frac{1}{1+2 \cos ^{5} x} d x $$

Perform the indicated integrations. $$ \int \frac{e^{\tan ^{-1} 2 t}}{1+4 t^{2}} d t $$

apply integration by parts twice to evaluate each integral. $$ \int x^{2} e^{x} d x $$

Use the method of partial fraction decomposition to perform the required integration. \(\int_{4}^{6} \frac{x-17}{x^{2}+x-12} d x\)

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. $$ \int_{-\pi / 4}^{\pi / 4} \frac{x^{3}}{4+\tan x} d x $$

Perform the indicated integrations. $$ \int(t+1) e^{-t^{2}-2 t-5} d t $$

apply integration by parts twice to evaluate each integral. $$ \int x^{5} e^{x^{2}} d x $$

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

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. $$ \int_{2}^{3} \frac{x^{2}+2 x-1}{x^{2}-2 x+1} d x $$

Perform the indicated integrations. $$ \int \frac{y}{\sqrt{16-9 y^{4}}} d y $$

apply integration by parts twice to evaluate each integral. $$ \int \ln ^{2} z d z $$

Use the method of partial fraction decomposition to perform the required integration. \(\int_{1}^{5} \frac{3 x+13}{x^{2}+4 x+3} d x\)

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. $$ \int_{1}^{3} \frac{d u}{u \sqrt{2 u-1}} $$

Perform the indicated integrations. $$ \int \cosh 3 x d x $$

apply integration by parts twice to evaluate each integral. $$ \int \ln ^{2} x^{20} d x $$

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

apply integration by parts twice to evaluate each integral. $$ \int e^{t} \cos t d t $$

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

Perform the indicated integrations. $$ \int \frac{5}{\sqrt{9-4 x^{2}}} d x $$

apply integration by parts twice to evaluate each integral. $$ \int e^{a t} \sin t d t $$

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

apply integration by parts twice to evaluate each integral. $$ \int x^{2} \cos x $$

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

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\() .\) $$ \delta(x)=\frac{x}{x^{2}+1} $$

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

apply integration by parts twice to evaluate each integral. $$ \int r^{2} \sin r d r $$

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

Perform the indicated integrations. $$ \int_{0}^{\pi / 2} \frac{\sin x}{16+\cos ^{2} x} d x $$

apply integration by parts twice to evaluate each integral. $$ \int \sin (\ln x) d x $$

Explain what happens to the solution of the logistic differential equation if the initial population size is larger than the maximum capacity.

Perform the indicated integrations. $$ \int_{0}^{1} \frac{e^{2 x}-e^{-2 x}}{e^{2 x}+e^{-2 x}} d x $$