Chapter 7

Solve each differential equation. $$ \frac{d y}{d x}+y=e^{-x} $$

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

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

In Problems 1-12, evaluate the given integral. $$ \int x e^{-5 x} d x $$

Perform the indicated integrations. $$ \int \sin ^{2} x d x $$

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

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

A slope field is given for a differential equation of the form \(y^{\prime}=f(x, y) .\) Use the slope field to sketch the solution that satisfies the given initial condition. In each case, find \(\lim _{x \rightarrow \infty} y(x)\) and approximate \(y(2) .\) $$ y(0)=6 $$

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

In Problems \(1-16\), perform the indicated integrations. \(\int x \sqrt[3]{x+\pi} d x\)

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

Evaluate the given integral. $$ \int \frac{x}{x^{2}+9} d x $$

Perform the indicated integrations. $$ \int \sin ^{4} 6 x d x $$

$$ \text { use integration by parts to evaluate each integral. } $$ $$ \int x e^{3 x} d x $$

Perform the indicated integrations. $$ \int \sqrt{3 x} d x $$

Solve each differential equation. $$ \left(1-x^{2}\right) \frac{d y}{d x}+x y=a x,|x|<1 $$

\(\int \frac{t d t}{\sqrt{3 t+4}}\)

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

Evaluate the given integral. $$ \int_{1}^{2} \frac{\ln x}{x} d x $$

Perform the indicated integrations. $$ \int \sin ^{3} x d x $$

Perform the indicated integrations. $$ \int_{0}^{2} x\left(x^{2}+1\right)^{5} d x $$

$$ \text { use integration by parts to evaluate each integral. } $$ $$ \int t e^{5 t+\pi} d t $$

A slope field is given for a differential equation of the form \(y^{\prime}=f(x, y) .\) Use the slope field to sketch the solution that satisfies the given initial condition. In each case, find \(\lim _{x \rightarrow \infty} y(x)\) and approximate \(y(2) .\) $$ y(1)=3 $$

Solve each differential equation. $$ y^{\prime}+y \tan x=\sec x $$

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

Use the method of partial fraction decomposition to perform the required integration. \(\int \frac{5 x}{2 x^{3}+6 x^{2}} d x\)

Evaluate the given integral. $$ \int \frac{x}{x^{2}-5 x+6} d x $$

Perform the indicated integrations. $$ \int \cos ^{3} x d x $$

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

$$ \text { use integration by parts to evaluate each integral. } $$ $$ \int(t+7) e^{2 t+3} d t $$

In Problems 1-16, perform the indicated integrations. \(\int_{1}^{2} \frac{d t}{\sqrt{t}+e}\)

Solve each differential equation. $$ \frac{d y}{d x}-\frac{y}{x}=x e^{x} $$

Use the method of partial fraction decomposition to perform the required integration. \(\int \frac{x-11}{x^{2}+3 x-4} d x\)

Evaluate the given integral. $$ \int \cos ^{4} 2 x d x $$

Perform the indicated integrations. $$ \int_{0}^{\pi / 2} \cos ^{5} \theta d \theta $$

Perform the indicated integrations. $$ \int \frac{d x}{x^{2}+4} $$

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

In Problems 1-16, perform the indicated integrations. \(\int_{0}^{1} \frac{\sqrt{t}}{t+1} d t\)

Use the method of partial fraction decomposition to perform the required integration. \(\int \frac{x-7}{x^{2}-x-12} d x\)

Evaluate the given integral. $$ \int \sin ^{3} x \cos x d x $$

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

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

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

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. $$ y^{\prime}=\frac{1}{2} y ; y(0)=\frac{1}{2} $$

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

Solve each differential equation. $$ \frac{d y}{d x}+\frac{y}{x}=\frac{1}{x} $$

Use the method of partial fraction decomposition to perform the required integration. \(\int \frac{3 x-13}{x^{2}+3 x-10} d x\)

Evaluate the given integral. $$ \int_{1}^{2} \frac{1}{x^{2}+6 x+8} d x $$

Perform the indicated integrations. $$ \int \sin ^{5} 4 x \cos ^{2} 4 x d x $$

Perform the indicated integrations. $$ \int \frac{x}{x^{2}+4} d x $$