Chapter 7

In Problems 1-36, use integration by parts to evaluate each integral. $$ \int(t-3) \cos (t-3) d t $$

In Problems 1-12, evaluate the given integral. $$ \int_{0}^{1 / 2} \frac{1}{1-t^{2}} d t $$

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

In Problems 7-10, 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 1-28, perform the indicated integrations. \(\int\left(\sin ^{3} 2 t\right) \sqrt{\cos 2 t} d t\)

\(\int x(1-x)^{2 / 3} d x\)

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

In Problems 1-14, solve each differential equation. $$ y^{\prime}+\frac{2 y}{x+1}=(x+1)^{3} $$

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

In Problems 1-12, evaluate the given integral. $$ \int_{0}^{5} x \sqrt{x+2} d x $$

In Problems 1-54, perform the indicated integrations. \(\int_{0}^{\sqrt{5}} 6 z \sqrt{4+z^{2}} d z\)

In Problems 7-10, 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. $$ y^{\prime}=x-y+2 ; y(0)=4 $$

In Problems 1-28, perform the indicated integrations. \(\int \cos ^{3} 3 \theta \sin ^{-2} 3 \theta d \theta\)

\(\int \frac{\sqrt{4-x^{2}}}{x} d x\)

In Problems 1–40, use the method of partial fraction decomposition to perform the required integration. $$ \int \frac{2 x+21}{2 x^{2}+9 x-5} d x $$

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

In Problems 1-12, evaluate the given integral. $$ \int_{3}^{4} \frac{1}{t-\sqrt{2 t}} d t $$

In Problems 1-54, perform the indicated integrations. \(\int_{0}^{4} \frac{5}{\sqrt{2 t+1}} d t\)

In Problems 1-28, perform the indicated integrations. \(\int \sin ^{1 / 2} 2 z \cos ^{3} 2 z d z\)

\(\int \frac{x^{2} d x}{\sqrt{16-x^{2}}}\)

In Problems 1–40, use the method of partial fraction decomposition to perform the required integration. $$ \int \frac{2 x^{2}-x-20}{x^{2}+x-6} d x $$

In Problems 1-14, solve each differential equation. $$ \frac{d y}{d x}+2 y=x $$

In Problems 1-36, use integration by parts to evaluate each integral. $$ \int t \sqrt[3]{2 t+7} d t $$

In Problems 1-12, evaluate the given integral. $$ \int_{-\pi / 2}^{\pi / 2} \cos ^{2} x \sin x d x $$

In Problems 1-54, perform the indicated integrations. \(\int_{0}^{\pi / 4} \frac{\tan z}{\cos ^{2} z} d z\)

In Problems 11-16, use Euler's Method with \(h=0.2\) to approximate the solution over the indicated interval. $$ y^{\prime}=2 y, y(0)=3,[0,1] $$

In Problems 1-28, perform the indicated integrations. \(\int \sin ^{4} 3 t \cos ^{4} 3 t d t\)

\(\int \frac{d x}{\left(x^{2}+4\right)^{3 / 2}}\)

In Problems 1–40, use the method of partial fraction decomposition to perform the required integration. $$ \int \frac{17 x-3}{3 x^{2}+x-2} d x $$

In Problems 1-14, solve each differential equation. $$ \frac{d y}{d x}-\frac{y}{x}=3 x^{3} ; y=3 \text { when } x=1 $$

In Problems 1-36, use integration by parts to evaluate each integral. $$ \int \ln 3 x d x $$

In Problems 1-12, evaluate the given integral. $$ \int_{0}^{2 \pi}|\sin 2 x| d x $$

In Problems 1-54, perform the indicated integrations. \(\int_{-\pi / 4}^{9 \pi / 4} e^{\cos z} \sin z d z\)

In Problems 11-16, use Euler's Method with \(h=0.2\) to approximate the solution over the indicated interval. $$ y^{\prime}=-y, y(0)=2,[0,1] $$

In Problems 1-28, perform the indicated integrations. \(\int \cos ^{6} \theta \sin ^{2} \theta d \theta\)

\(\int_{2}^{3} \frac{d t}{t^{2} \sqrt{t^{2}-1}}\)

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

In Problems 1-14, solve each differential equation. $$ y^{\prime}=e^{2 x}-3 y ; y=1 \text { when } x=0 $$

In Problems 1-36, use integration by parts to evaluate each integral. $$ \int \ln \left(7 x^{5}\right) d x $$

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

In Problems 11-16, use Euler's Method with \(h=0.2\) to approximate the solution over the indicated interval. $$ y^{\prime}=x, y(0)=0,[0,1] $$

In Problems 1-28, perform the indicated integrations. \(\int \sin 4 y \cos 5 y d y\)

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

In Problems 1-14, solve each differential equation. $$ x y^{\prime}+(1+x) y=e^{-x} ; y=0 \text { when } x=1 $$

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

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

In Problems 11-16, use Euler's Method with \(h=0.2\) to approximate the solution over the indicated interval. $$ y^{\prime}=x^{2}, y(0)=0,[0,1] $$

In Problems 1-28, perform the indicated integrations. \(\int \cos y \cos 4 y d y\)

\(\int \frac{t}{\sqrt{1-t^{2}}} d t\)

In Problems 1–40, use the method of partial fraction decomposition to perform the required integration. $$ \int \frac{7 x^{2}+2 x-3}{(2 x-1)(3 x+2)(x-3)} d x $$