Chapter 7

Evaluate the following integrals. $$\int_{0}^{5 / 2} \frac{d x}{\sqrt{25-x^{2}}}$$

Give the partial fraction decomposition for the following functions. $$\frac{5 x-7}{x^{2}-3 x+2}$$

How would you evaluate \(\int \tan ^{10} x \sec ^{2} x d x ?\)

Evaluate the following integrals. $$\int x \cos x d x$$

Evaluate the following integrals. $$\int \frac{d x}{(3-5 x)^{4}}$$

Explain how to sketch the direction field of the equation \(y^{\prime}(t)=F(t, y),\) where \(F\) is given.

Evaluate the following integrals or state that they diverge. $$\int_{1}^{\infty} 2^{-x} d x$$

Compute the absolute and relative errors in using c to approximate \(x\). \(x=\sqrt{2} ; c=1.414\)

Use a table of integrals to determine the following indefinite integrals. $$\int \frac{d x}{\sqrt{x^{2}-25}}$$

Evaluate the following integrals. $$\int_{0}^{3 / 2} \frac{d x}{\left(9-x^{2}\right)^{3 / 2}}$$

Give the partial fraction decomposition for the following functions. $$\frac{11 x-10}{x^{2}-x}$$

How would you evaluate \(\int \sec ^{12} x \tan x d x ?\)

Evaluate the following integrals. $$\int x \sin 2 x d x$$

Evaluate the following integrals. $$\int(9 x-2)^{-3} d x$$

Verify that the given function \(y\) is \(a\) solution of the differential equation that follows it. Assume that \(C, C_{1}\), and \(C_{2}\) are arbitrary constants. $$y=c e^{-5 t} ; y^{\prime}(t)+5 y=0$$

Compute the absolute and relative errors in using c to approximate \(x\). \(x=e ; c=2.72\)

Use a table of integrals to determine the following indefinite integrals. $$\int \frac{3 u}{2 u+7} d u$$

Evaluate the following integrals. $$\int_{5}^{10} \sqrt{100-x^{2}} d x$$

Give the partial fraction decomposition for the following functions. $$\frac{x^{2}}{x^{3}-16 x}, x \neq 0$$

Integrals of \(\sin x\) or \(\cos x\) Evaluate the following integrals. $$\int \sin ^{2} x d x$$

Evaluate the following integrals. $$\int t e^{t} d t$$

Evaluate the following integrals. $$\int_{0}^{3 \pi / 8} \sin \left(2 x-\frac{\pi}{4}\right) d x$$

Verify that the given function \(y\) is \(a\) solution of the differential equation that follows it. Assume that \(C, C_{1}\), and \(C_{2}\) are arbitrary constants. $$y=c t^{-3} ; t y^{\prime}(t)+3 y=0$$

Use a table of integrals to determine the following indefinite integrals. $$\int \frac{d y}{y(2 y+9)}$$

Evaluate the following integrals. $$\int_{0}^{\sqrt{2}} \frac{x^{2}}{\sqrt{4-x^{2}}} d x$$

Give the partial fraction decomposition for the following functions. $$\frac{x^{2}-3 x}{x^{3}-3 x^{2}-4 x}, x \neq 0$$

Integrals of \(\sin x\) or \(\cos x\) Evaluate the following integrals. $$\int \sin ^{3} x d x$$

Evaluate the following integrals. $$\int 2 x e^{3 x} d x$$

Evaluate the following integrals. $$\int e^{3-4 x} d x$$

Verify that the given function \(y\) is \(a\) solution of the differential equation that follows it. Assume that \(C, C_{1}\), and \(C_{2}\) are arbitrary constants. $$y=C_{1} \sin 4 t+C_{2} \cos 4 t ; y^{\prime \prime}(t)+16 y=0$$

Find the indicated Midpoint Rule approximations to the following integrals. \(\int_{2}^{10} 2 x^{2} d x\) using \(n=1,2,\) and 4 subintervals

Evaluate the following integrals or state that they diverge. $$\int_{0}^{\infty} e^{-2 x} d x$$

Use a table of integrals to determine the following indefinite integrals. $$\int \frac{d x}{1-\cos 4 x}$$

Evaluate the following integrals. $$\int_{0}^{1 / 2} \frac{x^{2}}{\sqrt{1-x^{2}}} d x$$

Give the partial fraction decomposition for the following functions. $$\frac{x+2}{x^{3}-3 x^{2}+2 x}$$

Integrals of \(\sin x\) or \(\cos x\) Evaluate the following integrals. $$\int \cos ^{3} x d x$$

Evaluate the following integrals. $$\int \frac{x}{\sqrt{x+1}} d x$$

Evaluate the following integrals. $$\int \frac{\ln 2 x}{x} d x$$

Verify that the given function \(y\) is \(a\) solution of the differential equation that follows it. Assume that \(C, C_{1}\), and \(C_{2}\) are arbitrary constants. $$y=C_{1} e^{-x}+C_{2} e^{x} ; y^{\prime \prime}(x)-y=0$$

Evaluate the following integrals or state that they diverge. $$\int_{4 / \pi}^{\infty} \frac{1}{x^{2}} \sec ^{2}\left(\frac{1}{x}\right) d x$$

Find the indicated Midpoint Rule approximations to the following integrals. \(\int_{1}^{9} x^{3} d x\) using \(n=1,2,\) and 4 subintervals

Use a table of integrals to determine the following indefinite integrals. $$\int \frac{d x}{x \sqrt{81-x^{2}}}$$

Evaluate the following integrals. $$\int_{1 / 2}^{1} \frac{\sqrt{1-x^{2}}}{x^{2}} d x$$

Give the partial fraction decomposition for the following functions. $$\frac{x^{2}-4 x+11}{(x-3)(x-1)(x+1)}$$

Integrals of \(\sin x\) or \(\cos x\) Evaluate the following integrals. $$\int \cos ^{4} 2 \theta d \theta$$

Evaluate the following integrals. $$\int s e^{-2 s} d s$$

Evaluate the following integrals. $$\int_{-5}^{0} \frac{d x}{\sqrt{4-x}}$$

Verify that the given function y is a solution of the initial value problem that follows it. $$y=16 e^{2 t}-10 ; y^{\prime}(t)-2 y=20, y(0)=6$$

Find the indicated Midpoint Rule approximations to the following integrals. \(\int_{0}^{1} \sin \pi x d x\) using \(n=6\) subintervals

Evaluate the following integrals or state that they diverge. $$\int_{0}^{\infty} e^{-\alpha x} d x, a>0$$