Chapter 6

Evaluate the given integral by making a trigonometric substitution (even if you spot another way to evaluate the integral). $$ \int \frac{1}{\sqrt{1+2 x^{2}}} d x $$

Integrate by parts to evaluate the given definite integral. $$ \int_{1}^{e} \ln (x) d x $$

Explicitly calculate the partial fraction decomposition of the given rational function. \(\frac{x^{3}+12 x^{2}-9 x+48}{(x-3)\left(x^{2}+4\right)}\)

Determine whether the given improper integral is convergent or divergent. If it converges, then evaluate it. \(\int_{0}^{3} x^{-1 / 2}(1+x) d x\)

Determine whether the given improper integral converges or diverges. If it converges, then evaluate it. $$ \int_{e}^{\infty} \frac{1}{x \ln (x)} d x $$

Evaluate the given definite integral. $$ \int_{0}^{\pi} \sin ^{3}(x) \cos ^{4}(x) d x $$

Decompose the given rational function into partial fractions. Calculate the coefficients. $$ \frac{2 x^{3}+x^{2}-5 x+2}{x^{2}(x+1)(x-2)} $$

Evaluate the given integral by making a trigonometric substitution (even if you spot another way to evaluate the integral). $$ \int \frac{1}{\left(1+x^{2}\right)^{3 / 2}} d x $$

Integrate by parts to evaluate the given definite integral. $$ \int_{1}^{e} x \ln (x) d x $$

Explicitly calculate the partial fraction decomposition of the given rational function. \(\frac{2 x^{2}+4 x+2}{\left(x^{2}+1\right)^{3}}\)

Determine whether the given improper integral is convergent or divergent. If it converges, then evaluate it. \(\int_{1}^{e} \frac{1}{x \cdot \ln ^{1 / 3}(x)} d x\)

Determine whether the given improper integral converges or diverges. If it converges, then evaluate it. $$ \int_{e}^{\infty} \frac{1}{x \ln ^{2}(x)} d x $$

Evaluate the given definite integral. $$ \int_{0}^{\pi / 2} \sin ^{2}(x) \cos ^{3}(x) d x $$

Decompose the given rational function into partial fractions. Calculate the coefficients. $$ \frac{1}{\left(x^{2}+x\right)^{2}} $$

Evaluate the given integral by making a trigonometric substitution (even if you spot another way to evaluate the integral). $$ \int \frac{1}{\left(5+x^{2}\right)^{3 / 2}} d x $$

Integrate by parts to evaluate the given definite integral. $$ \int_{1 / 3}^{1} x \ln (3 x) d x $$

Explicitly calculate the partial fraction decomposition of the given rational function. \(\frac{3 x^{3}-5 x^{2}+10 x-19}{\left(x^{2}+4\right)\left(x^{2}+3\right)}\)

Determine whether the given improper integral is convergent or divergent. If it converges, then evaluate it. \(\int_{0}^{1} \ln (x) d x\)

Determine whether the given improper integral converges or diverges. If it converges, then evaluate it. $$ \int_{1}^{\infty} \frac{\arctan (x)}{1+x^{2}} d x $$

Evaluate the given definite integral. $$ \int_{0}^{\pi} \sin ^{3}(x) \cos ^{6}(x) d x $$

Decompose the given rational function into partial fractions. Calculate the coefficients. $$ \frac{2 x^{3}-4 x^{2}-13 x+76}{\left(x^{2}-4 x+4\right)\left(x^{2}+6 x+9\right)} $$

Evaluate the given integral by making a trigonometric substitution (even if you spot another way to evaluate the integral). $$ \int \frac{2 x^{2}}{\sqrt{1+x^{2}}} d x $$

Integrate by parts to evaluate the given definite integral. $$ \int_{1}^{4} \frac{\ln (x)}{\sqrt{x}} d x $$

Explicitly calculate the partial fraction decomposition of the given rational function. \(\frac{2 x^{4}+15 x^{2}+30}{\left(x^{2}+4\right)\left(x^{2}+3\right)^{2}}\)

Determine whether the given improper integral is convergent or divergent. If it converges, then evaluate it. \(\int_{1}^{\sqrt{2}} \frac{1}{x \sqrt{x^{2}-1}} d x\)

Determine whether the given improper integral converges or diverges. If it converges, then evaluate it. $$ \int_{0}^{\infty}(2 / 3)^{x} d x $$

Evaluate the given definite integral. $$ \int_{0}^{\pi / 2} \sqrt{\sin (x)} \cos ^{3}(x) d x $$

Decompose the given rational function into partial fractions. Calculate the coefficients. $$ \frac{8}{(x-1)^{3}(x+1)} $$

Integrate by parts to evaluate the given definite integral. $$ \int_{0}^{1} x 3^{x} d x $$

Evaluate the given integral by making a trigonometric substitution (even if you spot another way to evaluate the integral). $$ \int \frac{1}{x \sqrt{1+x^{2}}} d x $$

In each of Exercises \(21-30\), use the method of partial fractions to decompose the integrand. Then evaluate the given integral. \(\int \frac{3 x^{2}-5 x+4}{(x-1)\left(x^{2}+1\right)} d x\)

In each of Exercises \(21-30\), determine whether the given improper integral is convergent or divergent. If it converges, then evaluate it. \(\int_{0}^{2} \frac{1}{x-1} d x\)

In each of Exercises \(21-36,\) determine whether the given improper integral converges or diverges. If it converges, then evaluate it. $$ \int_{-\infty}^{-2} x^{-3} d x $$

Evaluate the given definite integral. $$ \int_{0}^{\pi / 2} 15 \sin ^{2}(x) \cos ^{5}(x) d x $$

In each of Exercises \(21-26,\) use Heaviside's method to calculate the partial fraction decomposition of the given rational function. $$ \frac{x+1}{(2 x+7)(2 x+9)} $$

Integrate by parts successively to evaluate the given indefinite integral. $$ \int x^{2} e^{x} d x $$

Evaluate the given integral by making a trigonometric substitution (even if you spot another way to evaluate the integral). $$ \int_{2 \sqrt{2}}^{4} \frac{4}{x \sqrt{x^{2}-4}} d x $$

Use the method of partial fractions to decompose the integrand. Then evaluate the given integral. \(\int \frac{x^{2}+2}{\left(x^{2}+1\right) x^{2}} d x\)

Determine whether the given improper integral is convergent or divergent. If it converges, then evaluate it. \(\int_{-1}^{3} 6 x^{-1 / 7} d x\)

Determine whether the given improper integral converges or diverges. If it converges, then evaluate it. $$ \int_{-\infty}^{-2} x^{-1 / 3} d x $$

Evaluate the given definite integral. $$ \int_{0}^{3 \pi / 2} 5 \sin ^{2}(x / 3) \cos ^{7}(x / 3) d x $$

Use Heaviside's method to calculate the partial fraction decomposition of the given rational function. $$ \frac{2 x+1}{(x-2)(x+1)} $$

Integrate by parts successively to evaluate the given indefinite integral. $$ \int x^{2} e^{-x} d x $$

Evaluate the given integral by making a trigonometric substitution (even if you spot another way to evaluate the integral). $$ \int_{\sqrt{2}}^{2} \frac{1}{\sqrt{x^{2}-1}} d x $$

Use the method of partial fractions to decompose the integrand. Then evaluate the given integral. \(\int \frac{7 x^{3}+9 x-3 x^{2}-6}{\left(x^{2}+2\right)\left(x^{2}+1\right)} d x\)

Determine whether the given improper integral is convergent or divergent. If it converges, then evaluate it. \(\int_{-5}^{-1} \frac{3}{(x+3)^{2 / 5}} d x\)

Determine whether the given improper integral converges or diverges. If it converges, then evaluate it. $$ \int_{-\infty}^{-2} \frac{1}{(1+x)^{4 / 3}} d x $$

Use Heaviside's method to calculate the partial fraction decomposition of the given rational function. $$ \frac{5 x^{2}+3 x+1}{(x-2)(x+3)(x+4)} $$

Integrate by parts successively to evaluate the given indefinite integral. $$ \int x^{2} \cos (x) d x $$

Evaluate the given integral by making a trigonometric substitution (even if you spot another way to evaluate the integral). $$ \int_{1 / \sqrt{2}}^{\sqrt{5} / 2} \frac{2}{\sqrt{4 x^{2}-1}} d x $$