Chapter 7

$$ \text { use integration by parts to evaluate each integral. } $$ $$ \int(t-3) \cos (t-3) d t $$

Plot a slope field for each differential equation. Use the method of separation of variables (Section 4.9) or an integrating factor (Section 7.7) to find a particular solution of the differential equation that satisfies the given initial condition, and plot the particular solution. $$ y^{\prime}=-y ; y(0)=4 $$

In Problems 1-16, perform the indicated integrations. \(\int x(1-x)^{2 / 3} d x\)

Solve each differential equation. $$ y^{\prime}+\frac{2 y}{x+1}=(x+1)^{3} $$

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

Evaluate the given integral. $$ \int_{0}^{1 / 2} \frac{1}{1-t^{2}} d t $$

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

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

$$ \text { use integration by parts to evaluate each integral. } $$ $$ \int(x-\pi) \sin x d x $$

In Problems 1-16, perform the indicated integrations. \(\int \frac{\sqrt{4-x^{2}}}{x} d x\)

Plot a slope field for each differential equation. Use the method of separation of variables (Section 4.9) or an integrating factor (Section 7.7) to find a particular solution of the differential equation that satisfies the given initial condition, and plot the particular solution. $$ y^{\prime}=x-y+2 ; y(0)=4 $$

Use the method of partial fraction decomposition to perform the required integration. \(\int \frac{2 x+21}{2 x^{2}+9 x-5} d x\)

Evaluate the given integral. $$ \int_{0}^{5} x \sqrt{x+2} d x $$

Perform the indicated integrations. $$ \int \cos ^{3} 3 \theta \sin ^{-2} 3 \theta d \theta $$

Perform the indicated integrations. $$ \int_{0}^{\sqrt{5}} 6 z \sqrt{4+z^{2}} d z $$

$$ \text { use integration by parts to evaluate each integral. } $$ $$ \int t \sqrt{t+1} d t $$

In Problems 1-16, perform the indicated integrations. \(\int \frac{x^{2} d x}{\sqrt{16-x^{2}}}\)

Plot a slope field for each differential equation. Use the method of separation of variables (Section 4.9) or an integrating factor (Section 7.7) to find a particular solution of the differential equation that satisfies the given initial condition, and plot the particular solution. $$ y^{\prime}=2 x-y+\frac{3}{2} ; y(0)=3 $$

Solve each differential equation. $$ \frac{d y}{d x}+2 y=x $$

Use the method of partial fraction decomposition to perform the required integration. \(\int \frac{2 x^{2}-x-20}{x^{2}+x-6} d x\)

Evaluate the given integral. $$ \int_{3}^{4} \frac{1}{t-\sqrt{2 t}} d t $$

Perform the indicated integrations. $$ \int \sin ^{1 / 2} 2 z \cos ^{3} 2 z d z $$

Perform the indicated integrations. $$ \int_{0}^{4} \frac{5}{\sqrt{2 t+1}} d t $$

$$ \text { use integration by parts to evaluate each integral. } $$ $$ \int t \sqrt[3]{2 t+7} d t $$

In Problems 1-16, perform the indicated integrations. \(\int \frac{d x}{\left(x^{2}+4\right)^{3 / 2}}\)

Use Euler's Method with \(h=0.2\) to approximate the solution over the indicated interval. $$ y^{\prime}=2 y, y(0)=3,[0,1] $$

Solve each differential equation. $$ \frac{d y}{d x}-\frac{y}{x}=3 x^{3} ; y=3 \text { when } x=1 $$

Use the method of partial fraction decomposition to perform the required integration. \(\int \frac{17 x-3}{3 x^{2}+x-2} d x\)

Evaluate the given integral. $$ \int_{-\pi / 2}^{\pi / 2} \cos ^{2} x \sin x d x $$

Perform the indicated integrations. $$ \int \sin ^{4} 3 t \cos ^{4} 3 t d t $$

Perform the indicated integrations. $$ \int_{0}^{\pi / 4} \frac{\tan z}{\cos ^{2} z} d z $$

Perform the indicated integrations. \(\int_{2}^{3} \frac{d t}{t^{2} \sqrt{t^{2}-1}}\)

Use Euler's Method with \(h=0.2\) to approximate the solution over the indicated interval. $$ y^{\prime}=-y, y(0)=2,[0,1] $$

Solve each differential equation. $$ y^{\prime}=e^{2 x}-3 y ; y=1 \text { when } x=0 $$

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

Evaluate the given integral. $$ \int_{0}^{2 \pi}|\sin 2 x| d x $$

Perform the indicated integrations. $$ \int \cos ^{6} \theta \sin ^{2} \theta d \theta $$

Perform the indicated integrations. $$ \int_{-\pi / 4}^{9 \pi / 4} e^{\cos z} \sin z d z $$

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

Perform the indicated integrations. \(\int_{-2}^{-3} \frac{\sqrt{t^{2}}-1}{t^{3}} d t\)

Use Euler's Method with \(h=0.2\) to approximate the solution over the indicated interval. $$ y^{\prime}=x, y(0)=0,[0,1] $$

Solve each differential equation. $$ x y^{\prime}+(1+x) y=e^{-x} ; y=0 \text { when } x=1 \text { . } $$

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

Perform the indicated integrations. $$ \int \sin 4 y \cos 5 y d y $$

Perform the indicated integrations. $$ \int \frac{\sin \sqrt{t}}{\sqrt{t}} d t $$

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

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

Use Euler's Method with \(h=0.2\) to approximate the solution over the indicated interval. $$ y^{\prime}=x^{2}, y(0)=0,[0,1] $$

Solve each differential equation. $$ \sin x \frac{d y}{d x}+2 y \cos x=\sin 2 x ; y=2 \text { when } x=\frac{\pi}{6} $$

Use the method of partial fraction decomposition to perform the required integration. \(\int \frac{7 x^{2}+2 x-3}{(2 x-1)(3 x+2)(x-3)} d x\)