Chapter 29

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

In Problems 1 through 5, pinpoint all the improprieties in the integral. If necessary rewrite the integral as a sum of integrals so that each impropriety occurs at an endpoint and there is only one impropriety per integral. (a) \(\int_{0}^{\infty} \frac{1}{x^{2}+4} d x\) (b) \(\int_{0}^{\infty} \frac{1}{x^{2}-4} d x\)

Write out the partial fraction decomposition of each rational function. You need not determine the coefficients; just set them up. (a) \(\frac{x^{2}+3}{x(x-1)(x+5)}\) (b) \(\frac{x}{x^{3}+x}\)

In Problems 1 through 18 , evaluate the integral. \(\int \cos ^{2} x d x\)

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

Pinpoint all the improprieties in the integral. If necessary rewrite the integral as a sum of integrals so that each impropriety occurs at an endpoint and there is only one impropriety per integral. (a) \(\int_{0}^{\infty} \frac{1}{x^{2}} d x\) (b) \(\int_{-\infty}^{\infty} \frac{1}{x^{2}} d x\)

Write out the partial fraction decomposition of each rational function. You need not determine the coefficients; just set them up. (a) \(\frac{3}{x^{3}-4 x}\) (b) \(\frac{4}{x^{3}+2 x}\)

Evaluate the integral. \(\int_{0}^{\pi} \sin ^{3} x d x\)

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

Pinpoint all the improprieties in the integral. If necessary rewrite the integral as a sum of integrals so that each impropriety occurs at an endpoint and there is only one impropriety per integral. (a) \(\int_{-\infty}^{\infty} \frac{1}{x^{2}+4} d x\) (b) \(\int_{-\infty}^{\infty} \frac{1}{x^{2}-4} d x\)

Write out the partial fraction decomposition of each rational function. You need not determine the coefficients; just set them up. (a) \(\frac{x^{3}}{x^{3}-4 x}\) (b) \(\frac{3 x+1}{x^{4}+2 x^{2}+1}\)

Evaluate the integral. \(\int \cos x \sin ^{2} x d x\)

Evaluate the integrals. $$ \int_{1}^{e} \ln x d x $$

Write out the partial fraction decomposition of each rational function. You need not determine the coefficients; just set them up. (a) \(\frac{x+5}{x^{2}+3 x-4}\) (b) \(\frac{x+5}{x^{2}-4}\)

Evaluate the integral. \(\int \cos ^{3} x \sin ^{2} x d x\)

Evaluate the integrals. $$ \int_{0}^{1} \cos ^{-1} x d x $$

Write out the partial fraction decomposition of each rational function. You need not determine the coefficients; just set them up. $$\frac{3}{x^{3}+4 x}$$

Evaluate the integral. \(\int \cos ^{4} x d x\)

Evaluate the integrals. $$ \int \sin ^{-1}\left(\frac{x}{2}\right) d x $$

Show that \(\int_{1}^{\infty} \frac{1}{x^{p}} d x\) converges for \(p>1\) and diverges for \(p \leq 1\).

Write out the partial fraction decomposition of each rational function. You need not determine the coefficients; just set them up. \(\frac{x^{2}+1}{\left(x^{2}+x+1\right)(x-1)}\)

Evaluate the integral. \(\int \cos ^{2} x \sin ^{2} x d x\)

Evaluate the integrals. $$ \int e^{-t} \sin (2 t) d t $$

Show that \(\int_{0}^{1} \frac{1}{x^{p}} d x\) converges for \(p<1\) and diverges for \(p \geq 1\).

Evaluate the integrals. $$ \frac{3 x+9}{x^{2}-6 x+5} d x $$

Evaluate the integral. \(\int \cos ^{4} x \sin ^{3} x d x\)

Evaluate the integrals. $$ \int e^{-x} \cos x d x $$

Show that \(\int_{-1}^{\infty} \frac{1}{x^{4}} d x\) diverges.

Evaluate the integrals. $$ \int \frac{2}{x(x-1)^{2}} d x $$

Evaluate the integral. \(\int \cos ^{3} x \sin ^{11} x d x\)

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

(a) Evaluate \(\int_{0}^{\infty} x e^{-x^{2}} d x\). (b) Evaluate \(\int_{-\infty}^{\infty} x e^{-x^{2}} d x\).

Evaluate the integrals. $$ \int_{0}^{1} \frac{x^{2}}{2 x+3} d x $$

Evaluate the integral. \(\int \cos ^{3}(3 x) d x\)

Evaluate the integrals. $$ \int x \ln \frac{1}{x} d x $$

Show \(\int_{4}^{\infty} e^{-x^{2}} d x<0.0000001\). Hint : Compare it to \(\int_{4}^{\infty} x e^{-x^{2}} d x\).

Evaluate the integrals. $$ \int \frac{2}{x^{4}-1} d x $$

Evaluate the integral. \(\int_{0}^{\frac{\pi}{2}} \cos ^{5} x \sqrt{\sin x} d x\)

Evaluate the integrals. $$ \int x \sec ^{2} x d x $$

In Problems 11 through 36, determine whether the integral is convergent or divergent. Evaluate all convergent integrals. Be efficient. If \(\lim _{x \rightarrow \infty} \neq 0\), then \(\int_{a}^{\infty} f(x) d x\) is divergent. \(\int_{0}^{\infty} e^{-2 x} d x\)

Evaluate the integrals. $$ \int \frac{3 x^{2}+3}{\left(x^{2}-1\right)(x-2)} d x $$

Evaluate the integral. \(\int \frac{\sin x}{\sqrt{\cos ^{3} x}} d x\)

Evaluate the integrals. $$ \int_{0}^{1} t^{3} e^{-t} d t $$

Determine whether the integral is convergent or divergent. Evaluate all convergent integrals. Be efficient. If \(\lim _{x \rightarrow \infty} \neq 0\), then \(\int_{a}^{\infty} f(x) d x\) is divergent. \(\int_{0}^{\infty} x d x\)

Evaluate the integrals. $$ \int \frac{1}{x(x-3)} d x $$

Evaluate the integral. \(\int \tan 3 x d x\)

Evaluate the integrals. $$ \int \sqrt{x} \ln x d x $$

Evaluate the integrals. $$ \int \frac{e^{2 x}}{\left(e^{x}+2\right)\left(e^{x}-1\right)^{2}} d x $$

Evaluate the integral. \(\int \tan 2 x \sec 2 x d x\)

Evaluate the integrals. $$ \int \cos (\ln x) d x $$