Chapter 7

First make an appropriate substitution and then use integration by parts to evaluate the indefinite integrals. $$ \int \sin \sqrt{x} d x $$

a, b, and \(c\) are constants and \(g(x)\) is a continuous function whose derivative \(g^{\prime}(x)\) is also continuous. Use substitution to evaluate the indefinite integrals. $$ \int g^{\prime}(x) \sin [g(x)] d x $$

Find a comparison function for each integrand and determine whether the integral is convergent. $$ \int_{1}^{\infty} \frac{1}{\sqrt{1+x^{6}}} d x $$

$$ \text { In Problems } , \text { evaluate each definite integral. } $$ $$ \int_{1}^{2} \frac{x^{2}+1}{x} d x $$

First make an appropriate substitution and then use integration by parts to evaluate the indefinite integrals. $$ \int x^{3} e^{-x^{2} / 2} d x $$

a, b, and \(c\) are constants and \(g(x)\) is a continuous function whose derivative \(g^{\prime}(x)\) is also continuous. Use substitution to evaluate the indefinite integrals. $$ \int g^{\prime}(x) e^{-g(x)} d x $$

Find a comparison function for each integrand and determine whether the integral is convergent. $$ \int_{1}^{\infty} \frac{1}{\sqrt{1+x}} d x $$

$$ \text { In Problems } , \text { evaluate each definite integral. } $$ $$ \int_{2}^{3} \frac{1}{1-x} d x $$

First make an appropriate substitution and then use integration by parts to evaluate the indefinite integrals. $$ \int x^{5} e^{x^{2}} d x $$

a, b, and \(c\) are constants and \(g(x)\) is a continuous function whose derivative \(g^{\prime}(x)\) is also continuous. Use substitution to evaluate the indefinite integrals. $$ \int \frac{g^{\prime}(x)}{[g(x)]^{2}+1} d x $$

Find a comparison function for each integrand and determine whether the integral is convergent. $$ \int_{-\infty}^{\infty} \frac{1}{e^{x}+e^{-x}} d x $$

$$ \text { In Problems } , \text { evaluate each definite integral. } $$ $$ \int_{2}^{3} \frac{1}{1-x^{2}} d x $$

First make an appropriate substitution and then use integration by parts to evaluate the indefinite integrals. $$ \int \sin x \cos x e^{\sin x} d x $$

In Problems 43-58, use substitution to evaluate the definite integrals. $$ \int_{0}^{3} x \sqrt{x^{2}+1} d x $$

(a) Show that $$ \lim _{x \rightarrow \infty} \frac{\ln x}{\sqrt{x}}=0 $$ (b) Use your result in (a) to show that $$ 2 \ln x \leq \sqrt{x} $$ for sufficiently large \(x .\) Use a graphing calculator to determine just how large \(x\) must be for \((7.17)\) to hold. (c) Use your result in (b) to show that $$ \int_{0}^{\infty} e^{-\sqrt{x}} d x $$ converges. Use a graphing calculator to sketch the function \(f(x)=e^{-\sqrt{x}}\) together with its comparison function(s), and use your graph to explain how you showed that the integral in (7.18) is convergent.

$$ \text { In Problems } , \text { evaluate each definite integral. } $$ $$ \int_{0}^{1} \tan ^{-1} x d x $$

First make an appropriate substitution and then use integration by parts to evaluate the indefinite integrals. $$ \int \sin x \cos ^{3} x e^{1-\sin ^{2} x} d x $$

(a) Show that $$ \lim _{x \rightarrow \infty} \frac{\ln x}{x}=0 $$ (b) Use your result in (a) to show that, for any \(c>0\), $$ c x \geq \ln x $$ for sufficiently large \(x\). (c) Use your result in (b) to show that, for any \(p>0\), $$ x^{p} e^{-x} \leq e^{-x / 2} $$ provided that \(x\) is sufficiently large. (d) Use your result in (c) to show that, for any \(p>0\), $$ \int_{0}^{\infty} x^{p} e^{-x} d x $$ is convergent.

$$ \text { In Problems } , \text { evaluate each definite integral. } $$ $$ \int_{0}^{1} x \tan ^{-1} x d x $$

First make an appropriate substitution and then use integration by parts to evaluate the indefinite integrals. $$ \int_{0}^{1} e^{\sqrt{x}} d x $$

Use substitution to evaluate the definite integrals. $$ \int_{2}^{3} \frac{2 x+3}{\left(x^{2}+3 x\right)^{3}} d x $$

$$ \text { In Problems 45-52, evaluate each integral. } $$ $$ \int \frac{1}{(x+1)^{2} x} d x $$

Use substitution to evaluate the definite integrals. $$ \int_{0}^{2} \frac{2 x}{\left(4 x^{2}+3\right)^{1 / 3}} d x $$

$$ \text { In Problems , evaluate each integral. } $$ $$ \int \frac{1}{x^{2}(x-1)^{2}} d x $$

Use substitution to evaluate the definite integrals. $$ \int_{2}^{5}(x-2) e^{-(x-2)^{2} / 2} d x $$

$$ \text { In Problems , evaluate each integral. } $$ $$ \int \frac{4}{(1-x)(1+x)^{2}} d x $$

First make an appropriate substitution and then use integration by parts to evaluate the indefinite integrals. $$ \int_{0}^{1} x^{3} \ln \left(x^{2}+1\right) d x $$

Use substitution to evaluate the definite integrals. $$ \int_{\ln 4}^{\ln 7} \frac{e^{x}}{\left(e^{x}-3\right)^{2}} d x $$

$$ \text { In Problems , evaluate each integral. } $$ $$ \int \frac{2 x^{2}+2 x-1}{x^{3}(x-3)} d x $$

Use either substitution or integration by parts to evaluate each integral. $$ \int x e^{-2 x} d x $$

Use substitution to evaluate the definite integrals. $$ \int_{0}^{\pi / 3} \sin x \cos x d x $$

$$ \text { In Problems , evaluate each integral. } $$ $$ \int \frac{1}{\left(x^{2}-9\right)^{2}} d x $$

Use either substitution or integration by parts to evaluate each integral. $$ \int x e^{-2 x^{2}} d x $$

Use substitution to evaluate the definite integrals. $$ \int_{-\pi / 6}^{\pi / 6} \sin ^{2} x \cos x d x $$

$$ \text { In Problems , evaluate each integral. } $$ $$ \int \frac{1}{\left(x^{2}-x-2\right)^{2}} d x $$

Use either substitution or integration by parts to evaluate each integral. $$ \int \frac{1}{\tan x} d x $$

Use substitution to evaluate the definite integrals. $$ \int_{0}^{\pi / 4} \tan x \sec ^{2} x d x $$

$$ \text { In Problems , evaluate each integral. } $$ $$ \int \frac{1}{x^{2}\left(x^{2}+1\right)} d x $$

Use either substitution or integration by parts to evaluate each integral. $$ \int \frac{1}{\csc x \sec x} d x $$

Use substitution to evaluate the definite integrals. $$ \int_{0}^{\pi / 3} \frac{\sin x}{\cos ^{2} x} d x $$

$$ \text { In Problems , evaluate each integral. } $$ $$ \int \frac{1}{(x+1)^{2}\left(x^{2}+1\right)} d x $$

Use either substitution or integration by parts to evaluate each integral. $$ \int 2 x \sin \left(x^{2}\right) d x $$

Use substitution to evaluate the definite integrals. $$ \int_{5}^{9} \frac{x}{x-3} d x $$

Use either substitution or integration by parts to evaluate each integral. $$ \int 2 x^{2} \sin x d x $$

Use substitution to evaluate the definite integrals. $$ \int_{0}^{2} \frac{x}{x+2} d x $$

Use either substitution or integration by parts to evaluate each integral. $$ \int \frac{1}{16+x^{2}} d x $$

Use substitution to evaluate the definite integrals. $$ \int_{e}^{e^{2}} \frac{d x}{x(\ln x)^{2}} $$

Use either substitution or integration by parts to evaluate each integral. $$ \int \frac{1}{x^{2}+5} d x $$

Use substitution to evaluate the definite integrals. $$ \int_{1}^{2} \frac{x d x}{\left(x^{2}+1\right) \ln \left(x^{2}+1\right)} $$

Use either substitution or integration by parts to evaluate each integral. $$ \int \frac{x}{x+3} d x $$