Chapter 6

In each of Exercises \(1-10,\) determine whether the given improper integral is convergent or divergent. If it converges, then evaluate it. \(\int_{1}^{5}(x-5)^{-4 / 3} d x\)

In each of Exercises \(1-20\), determine whether the given improper integral converges or diverges. If it converges, then evaluate it. $$ \int_{3}^{\infty} x^{-3 / 2} d x $$

In each of Exercises \(1-10,\) write down the form of the partial fraction decomposition of the given rational function. Do not explicitly calculate the coefficients. \(\frac{2 x^{3}+x+1}{\left(x^{2}+1\right)\left(x^{2}+4\right)}\)

Evaluate the given integral by making a trigonometric substitution (even if you spot another way to evaluate the integral). $$ \int \frac{x}{\sqrt{9-x^{2}}} d x $$

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

Evaluate the given integral. $$ \int \sin ^{2}(x / 2) d x $$

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

Determine whether the given improper integral converges or diverges. If it converges, then evaluate it. $$ \int_{9}^{\infty} \frac{1}{\sqrt{x}} d x $$

Write down the form of the partial fraction decomposition of the given rational function. Do not explicitly calculate the coefficients. \(\frac{2 x^{4}}{\left(x^{2}+2 x+2\right)\left(2 x^{2}+5 x+3\right)}\)

Evaluate the given integral by making a trigonometric substitution (even if you spot another way to evaluate the integral). $$ \int \frac{1}{\sqrt{9-x^{2}}} d x $$

Evaluate the given integral. $$ \int \sin ^{2}(4 x) d x $$

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

Determine whether the given improper integral is convergent or divergent. If it converges, then evaluate it. \(\int_{2}^{4}(4-x)^{-0.9} d x\)

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

Write down the form of the partial fraction decomposition of the given rational function. Do not explicitly calculate the coefficients. \(\frac{2 x^{3}+x+1}{\left(x^{2}+1\right)\left(x^{2}+x+1\right)^{2}}\)

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

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

Integrate by parts to evaluate the given indefinite integral. $$ \int(2 x+5) e^{x / 3} d x $$

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

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

Write down the form of the partial fraction decomposition of the given rational function. Do not explicitly calculate the coefficients. \(\frac{2 x^{4}}{\left(x^{2}+1\right)\left(5 x^{2}+4 x+1\right)}\)

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

Evaluate the given integral. $$ \int \cos ^{2}(x+\pi / 3) d x $$

Integrate by parts to evaluate the given indefinite integral. $$ \int x \sin (x) d x $$

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

Determine whether the given improper integral converges or diverges. If it converges, then evaluate it. $$ \int_{0}^{\infty} \frac{1}{1+x^{2}} d x $$

Write down the form of the partial fraction decomposition of the given rational function. Do not explicitly calculate the coefficients. \(\frac{2 x+1}{\left(x^{2}+x+3\right)(x-4)}\)

Evaluate the given integral. $$ \int \tan ^{2}(x / 2) d x $$

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

Integrate by parts to evaluate the given indefinite integral. $$ \int(4 x+2) \sin (2 x) d x $$

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

Determine whether the given improper integral converges or diverges. If it converges, then evaluate it. $$ \int_{0}^{\infty} \frac{x}{1+x^{2}} d x $$

Write down the form of the partial fraction decomposition of the given rational function. Do not explicitly calculate the coefficients. \(\frac{3 x^{3}+x+1}{x^{2}(x+1)^{2}}\)

Evaluate the given integral. $$ \int\left(\cos ^{2}(x)-\sin ^{2}(x)\right) d x $$

Evaluate the given integral by making a trigonometric substitution (even if you spot another way to evaluate the integral). $$ \int \sqrt{25-x^{2}} d x $$

Integrate by parts to evaluate the given indefinite integral. $$ \int 9 x \cos (3 x) d x $$

Write down the form of the partial fraction decomposition of the given rational function. Do not explicitly calculate the coefficients. \(\frac{2 x^{6}}{\left(x^{2}+4\right)^{3}(x-2)}\)

Determine whether the given improper integral is convergent or divergent. If it converges, then evaluate it. \(\int_{0}^{1} \frac{x}{\left(1-x^{2}\right)^{1 / 4}} d x\)

Determine whether the given improper integral converges or diverges. If it converges, then evaluate it. $$ \int_{0}^{\infty} \frac{x}{\left(1+x^{2}\right)^{2}} d x $$

Evaluate the given indefinite integral. $$ \int 6 \sin ^{3}(2 x) d x $$

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

Integrate by parts to evaluate the given indefinite integral. $$ \int x \ln (4 x) d x $$

Write down the form of the partial fraction decomposition of the given rational function. Do not explicitly calculate the coefficients. \(\frac{3 x^{5}+x+1}{x^{2}\left(x^{2}-1\right)^{2}}\)

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

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

Evaluate the given indefinite integral. $$ \int 3 \cos ^{3}(x+2) d x $$

Evaluate the given integral by making a trigonometric substitution (even if you spot another way to evaluate the integral). $$ \int\left(4-x^{2}\right)^{3 / 2} d x $$

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

Write down the form of the partial fraction decomposition of the given rational function. Do not explicitly calculate the coefficients. \(\frac{3 x^{5}+x+1}{x^{2}\left(x^{2}-1\right)\left(x^{2}+1\right)}\)

Determine whether the given improper integral is convergent or divergent. If it converges, then evaluate it. \(\int_{0}^{2} \frac{1}{4-x^{2}} d x\)