Chapter 25

For Problems 1 through 11, find the given indefinite integral. $$ \int \frac{x}{x+1} d x $$

Evaluate the following indefinite integrals. (Let your mind be limber. Integrals that look very similar may require very different mindsets!) (a) \(\int \frac{1}{x} d x\) (b) \(\int \frac{1}{x+1} d x\) (c) \(\int \frac{1}{(x+1)^{2}} d x\) (d) \(\int \frac{1}{x^{2}+1} d x\) (e) \(\int \frac{x}{x^{2}+1} d x\) (f) \(\int \frac{x^{2}+1}{x} d x\) (g) \(\int(1+x)^{5} d x\) (h) \(\int \frac{1}{(1+x)^{5}} d x\) (i) \(\int\left(1+x^{2}\right)^{2} d x\)

In Problems 1 through 11, compute the integral. \(\int 3 x^{3}+2 x+\pi d x\)

Find the given indefinite integral. $$ \int \frac{x+3}{x-7} d x $$

Find the following indefinite integrals. Check your answers. (a) \(\int 3 \sin (5 t) d t\) (b) \(\int \pi \cos (\pi t) d t\) (c) \(\int \sqrt{3 x+5} d x\) (d) \(\int \frac{\pi}{e^{x}} d x\) (e) \(\int e^{-3 t} d t\) (f) \(\int \sqrt{e^{t}} d t\) (g) \(\int \frac{6}{\sqrt{t^{3}}} d t\) (h) \(\int \frac{1}{3 t+8} d t\)

Compute the integral. \(\int A t^{n} d t\), where \(A\) and \(n\) are constants and \(n \neq-1 .\)

Find the given indefinite integral. $$ \int x \sqrt{3 x+5} d x $$

In Problems 3 through 8, nd the inde nite integrals. (a) \(\int(2 x+1)^{3} d x\) (b) \(\int \frac{1}{(2 x+1)^{2}} d x\) (c) \(\int \frac{1}{(2 x+1)} d x\) (d) \(\int \frac{1}{\sqrt{2 x+1}} d x\)

Compute the integral. \(\int 3 x^{-1} d x\)

Find the given indefinite integral. $$ \frac{2 x}{3+x} d x $$

nd the inde nite integrals. (a) \(\int x \sqrt{2 x^{2}+1} d x\) (b) \(\int \frac{x}{\sqrt{2 x^{2}+1}} d x\) (c) \(\int \frac{\cos \sqrt{x}}{\sqrt{x}} d x\) (d) \(\int \sqrt{\cos x} \sin x d x\)

Compute the integral. \(\int \frac{d x}{2 x}\)

Find the given indefinite integral. $$ \int 3 t \sqrt{t^{2}+5} d t $$

nd the inde nite integrals. (a) \(\int \frac{5}{1+9 x^{2}} d x\) (b) \(\int \frac{\ln x}{x} d x\) (c) \(\int \frac{e^{x}}{e^{-x}} d x\) (d) \(\int \frac{(\ln w)^{2}}{w} d w\)

Compute the integral. \(\int 3 \sin t-\frac{3}{1+t^{2}} d t\)

Find the given indefinite integral. $$ \int t \sqrt{2 t+5} d t $$

nd the inde nite integrals. (a) \(\int t^{2} \sin \left(t^{3}\right) d t\) (b) \(\int x e^{-x^{2}} d t\) (c) \(\int \frac{1}{x+5} d x\) (d) \(\int \frac{t}{2 t^{2}+7} d t\)

Compute the integral. \(\int \frac{5 d x}{7 x}\)

Find the given indefinite integral. $$ \int \frac{x^{\pi}}{5} d x $$

nd the inde nite integrals. (a) \(\int \frac{t}{\left(t^{2}+1\right)^{2}} d t\) (b) \(\int \tan 2 t d t\) (c) \(\int \frac{e^{w}}{e^{w}+1} d w\) (d) \(\int \frac{e^{w}}{e^{2 w}+1} d w\)

Compute the integral. \(\int \frac{2 \cos w}{3} d w\)

Find the given indefinite integral. $$ \cot (3 x) d x $$

nd the inde nite integrals. (a) \(\int\left(x^{2}+3\right)^{3} x d x\) (b) \(\int x \sqrt{x^{2}+4} d x\) (c) \(\int(2 x+1) \sqrt{x^{2}+x} d x\)

Compute the integral. \(\int \frac{1}{x^{2}+1} d x\)

Find the given indefinite integral. $$ \int(1.5)^{(1-t)} d t $$

Evaluate. (a) \(\int_{0}^{1} \frac{3}{1+4 w^{2}} d w\) (b) \(\int_{0}^{1} \frac{1+4 w^{2}}{3} d w\) (c) \(\int_{\pi / 2}^{3 \pi} \cos \left(\frac{t}{2}\right) d t\) (d) \(\int_{1}^{3} \frac{4}{3 x+2} d x\) (e) \(\int_{1}^{4} \frac{1}{(2 x+1)^{2}} d x\)

Compute the integral. \(\int \frac{e^{p}}{2} d p\)

Find the given indefinite integral. $$ \int \sin ^{4} t \cos t d t $$

Find the following definite integrals. (a) \(\int_{1}^{3} \frac{1}{(2 x+4)^{2}}+\frac{1}{x+4} d x\) (b) \(\int_{1}^{e} \frac{(\ln x)^{2}}{x} d x\)

Compute the integral. \(\int \frac{\sec ^{2} t}{5} d t\)

Find the given indefinite integral. $$ \int \frac{2 t}{\sqrt{2 t+6}} d t $$

In Problems 11 through 23, compute the following integrals. $$ \int \frac{\cos \left(e^{-x}\right)}{e^{x}} d x $$

Compute the integral. \(\int \cos t+\sec t \tan t d t\)

Suppose you ve forgotten the antiderivative of \(\frac{1}{\sqrt{1-x^{2}}} .\) In this problem you will use a sophisticated substitution that will help you proceed. The goal is to nd $$ \int_{0}^{1 / 2} \frac{1}{\sqrt{1-x^{2}}} d x $$

Compute the following integrals. $$ \int\left(e^{x}+x\right) \sqrt{2 e^{x}+x^{2}} d x $$

In Problems 12 through 14, nd antiderivatives for the given functions. In other words, for each function \(f, \quad\) nd a function \(F\) such that \(F^{\prime}=f .\) Check your answers. (a) \(f(x)=e^{3 x}\) (b) \(f(x)=\frac{3}{e^{x}}\)

Evaluate \(\int \frac{2}{x(x+2)} d x\) by rewriting the integrand in the form $$ \frac{A}{x}+\frac{B}{x+2} $$ where \(A\) and \(B\) are constants. In other words, nd \(A\) and \(B\) such that $$ \frac{2}{x(x+2)}=\frac{A}{x}+\frac{B}{x+2} $$ (This is an example of an integration technique known as partial fractions.)

Compute the following integrals. $$ \int \frac{\sec ^{2}(\ln x)}{x} d x $$

Nd antiderivatives for the given functions. In other words, for each function \(f, \quad\) nd a function \(F\) such that \(F^{\prime}=f .\) (a) \(f(x)=\frac{-1}{2 x}\) (b) \(f(x)=\frac{4}{1+x^{2}}\)

Evaluate \(\int \frac{1}{x^{2}-1} d x\) by factoring the denominator of the integrand and rewriting the integrand in the form $$ \frac{A}{x-1}+\frac{B}{x+1} $$ where \(A\) and \(B\) are constants.

Compute the following integrals. $$ \int \frac{\sin (x)}{\sqrt{\cos (x)}} d x $$

Nd antiderivatives for the given functions. In other words, for each function \(f, \quad\) nd a function \(F\) such that \(F^{\prime}=f .\) (a) \(f(x)=\sin 2 x\) (b) \(f(x)=\cos (x / 3)\)

Compute the following integrals. $$ \int_{0}^{\pi / 4} \tan x d x $$

(a) Differentiate \(f(x)=5 \tan \left(x^{2}\right)+\arctan 3 x\). (b) Find \(\int 10 x \sec ^{2}\left(x^{2}\right)+\frac{3}{1+9 x^{2}} d x\)

Compute the following integrals. $$ \int e^{2 x} \sqrt{e^{x}} d x $$

(a) Differentiate \(y=\frac{-\pi \cos 3 x}{3}\). (b) Find \(\int A \sin B x d x\), where \(A\) and \(B\) are constants.

Compute the following integrals. $$ \int_{0}^{\pi / 2} \frac{\cos (x)}{1+\sin ^{2} x} d x $$

(a) Suppose the velocity of an object is given by \(v(t)=\frac{1}{1+t^{2}}\) miles per hour. Find its net change in position from \(t=0\) to \(t=1, t\) measured in hours. What is the total distance traveled on \([0,1] ?\) (b) Suppose that the velocity of an object is given by \(v(t)=5 t(t-1)\) meters per second. What is the net change in position between \(t=0\) and \(t=2, t\) measured in seconds? What is the total distance traveled on \([0,2] ?\)

Compute the following integrals. $$ \int_{0}^{\ln 5} \frac{3 e^{x}}{\sqrt{e^{x}+4}} d x $$

Find the following indefinite integrals. (a) \(\int \frac{2+x}{x} d x\) (b) \(\int \frac{3}{x^{2}} d x\) (c) \(\int \frac{3}{1+x^{2}} d x\) (d) \(\int\left(\frac{t^{3}}{4}+\frac{4}{\sqrt{t}}\right) d t\)