Using Two Methods In Exercises 1316, find the indefinite integral (a) using integration

In Exercises 1 and 2, use a table of integrals with forms involving $a+b u$ to find the indefinite integral. \(\int \frac{x^{2}}{5+x} d x\) Text Transcription: nt x^2 / 5 + x dx

In Exercises 1 and 2, use a table of integrals with forms involving $a+b u$ to find the indefinite integral. \(\int \frac{2}{x^{2}(4+3 x)^{2}} d x\) Text Transcription: int 2 / x^2(4 + 3x)^2 dx

In Exercises 3 and 4, use a table of integrals with forms involving \(\sqrt{a^{2}-u^{2}}\) to find the indefinite integral. \(\int \frac{1}{x^{2} \sqrt{1-x^{2}}} d x\) Text Transcription: sqrt{a^2 - u^2 int 1 / x^2 sqrt{1 - x^2} dx

In Exercises 3 and 4, use a table of integrals with forms involving \(\sqrt{a^{2}-u^{2}}\) to find the indefinite integral. \(\int \frac{\sqrt{64-x^{4}}}{x} d x\) Text Transcription: sqrt{a^2 - u^2} int sqrt{64 - x^4} x dx

In Exercises 5-8, use a table of integrals with forms involving the trigonometric functions to find the indefinite integral. \(\int \cos ^{4} 3 x d x\) Text Transcription: int cos^4 3x dx

In Exercises 5-8, use a table of integrals with forms involving the trigonometric functions to find the indefinite integral. \(\int \frac{\sin ^{4} \sqrt{x}}{\sqrt{x}} d x\) Text Transcription: int sin^{4} sqrt x / sqrt x dx

In Exercises 5-8, use a table of integrals with forms involving the trigonometric functions to find the indefinite integral. \(\int \frac{1}{\sqrt{x}(1-\cos \sqrt{x})} d x\) Text Transcription: int 1 / sqrt x (1 - cos sqrt x) dx

In Exercises 5-8, use a table of integrals with forms involving the trigonometric functions to find the indefinite integral. \(\int \frac{1}{1+\cot 4 x} d x\) Text Transcription: int 1 / 1 + cot 4x dx

In Exercises 9 and 10, use a table of integrals with forms involving $e^{u}$ to find the indefinite integral. \(\int \frac{1}{1+e^{2 x}} d x\) Text Transcription: int 1 / 1 + e^2x dx

In Exercises 9 and 10, use a table of integrals with forms involving $e^{u}$ to find the indefinite integral. \(\int e^{-4 x} \sin 3 x d x\) Text Transcription: int e^{-4x} sin 3x dx

In Exercises 11 and 12, use a table of integrals with forms involving \(\ln u\) to find the indefinite integral. \(\int x^{7} \ln x d x\) Text Transcription: ln u int x^{7} ln x dx

In Exercises 11 and 12, use a table of integrals with forms involving \(\ln u\) to find the indefinite integral. \(\int(\ln x)^{3} d x\) Text Transcription: ln u int(ln x)^3 dx

In Exercises 13-16, find the indefinite integral (a) using integration tables and (b) using the given method. Integral Method \(\int x^{2} e^{3 x} d x\) Integration by parts Text Transcription: int x^{2} e^{3x} dx

In Exercises 13-16, find the indefinite integral (a) using integration tables and (b) using the given method. Integral Method \(\int x^{5} \ln x d x\) Integration by parts Text Transcription: int x^{5} ln x dx

In Exercises 13-16, find the indefinite integral (a) using integration tables and (b) using the given method. Integral Method \(\int \frac{1}{x^{2}(x+1)} d x\) Partial fractions Text Transcription: int 1 / x^2 (x + 1) dx

In Exercises 13-16, find the indefinite integral (a) using integration tables and (b) using the given method. Integral Method \(\int \frac{1}{x^{2}-36} d x\) Partial fractions Text Transcription: int 1 / x^2 - 36 dx

In Exercises 17-38, use integration tables to find the indefinite integral. \(\int x \operatorname{arccsc}\left(x^{2}+1\right) d x\) Text Transcription: int x arccsc (x^2 + 1) dx

In Exercises 17-38, use integration tables to find the indefinite integral. \(\int \arcsin 4 x d x\) Text Transcription: int arcsin 4x dx

In Exercises 17-38, use integration tables to find the indefinite integral. \(\int \frac{1}{x^{2} \sqrt{x^{2}-4}} d x\) Text Transcription: int 1 / x^2 sqrt{x^2 - 4} dx

In Exercises 17-38, use integration tables to find the indefinite integral. \(\int \frac{1}{x^{2}+4 x+8} d x\) Text Transcription: int 1 / x^2 + 4x + 8 dx

In Exercises 17-38, use integration tables to find the indefinite integral. \(\int \frac{4 x}{(2-5 x)^{2}} d x\) Text Transcription: int 4x / (2 - 5x)^2 dx

In Exercises 17-38, use integration tables to find the indefinite integral. \(\int \frac{\theta^{3}}{1+\sin \theta^{4}} d \theta\) Text Transcription: int theta^3 / 1 + sin theta^4 d theta

In Exercises 17-38, use integration tables to find the indefinite integral. \(\int e^{x} \arccos e^{x} d x\) Text Transcription: int e^{x} arccos e^x dx

In Exercises 17-38, use integration tables to find the indefinite integral. \(\int \frac{e^{x}}{1-\tan e^{x}} d x\) Text Transcription: int e^x / 1 - tan e^x dx

In Exercises 17-38, use integration tables to find the indefinite integral. \(\int \frac{x}{1-\sec x^{2}} d x\) Text Transcription: int x / 1 - sec x^2 dx

In Exercises 17-38, use integration tables to find the indefinite integral. \(\int \frac{1}{t\left[1+(\ln t)^{2}\right]} d t\) Text Transcription: int 1 / t[1 + (ln t)^2] dt

In Exercises 17-38, use integration tables to find the indefinite integral. \(\int \frac{\cos \theta}{3+2 \sin \theta+\sin ^{2} \theta} d \theta\) Text Transcription: int cos theta / 3 + 2 sin theta + sin ^2 theta d theta

In Exercises 17-38, use integration tables to find the indefinite integral. \(\int x^{2} \sqrt{2+9 x^{2}} d x\) Text Transcription: int x^{2} sqrt{2 + 9x^2} dx

In Exercises 17-38, use integration tables to find the indefinite integral. \(\int \frac{1}{x^{2} \sqrt{2+9 x^{2}}} d x\) Text Transcription: int 1 / x^{2} sqrt 2 + 9x^2 dx

In Exercises 17-38, use integration tables to find the indefinite integral. \(\int \sqrt{x} \arctan x^{3 / 2} d x\) Text Transcription: int sqrt{x} arctan x^{3/2} dx

In Exercises 17-38, use integration tables to find the indefinite integral. \(\int \frac{\ln x}{x(3+2 \ln x)} d x\) Text Transcription: int ln x / x(3 + 2 ln x) dx

In Exercises 17-38, use integration tables to find the indefinite integral. \(\int \frac{e^{x}}{\left(1-e^{2 x}\right)^{3 / 2}} d x\) Text Transcription: int e^x / (1 - e^{2x})^{3/2} dx

In Exercises 17-38, use integration tables to find the indefinite integral. \(\int \frac{x}{\left(x^{2}-6 x+10\right)^{2}} d x\) Text Transcription: int x / (x^{2} - 6x + 10)^{2} dx

In Exercises 17-38, use integration tables to find the indefinite integral. \(\int \sqrt{\frac{5-x}{5+x}} d x\) Text Transcription: int sqrt{5 - x / 5 + x} dx

In Exercises 17-38, use integration tables to find the indefinite integral. \(\int \frac{x}{\sqrt{x^{4}-6 x^{2}+5}} d x\) Text Transcription: int x / sqrt{x^4 - 6x^2 + 5} dx

In Exercises 17-38, use integration tables to find the indefinite integral. \(int \frac{\cos x}{\sqrt{\sin ^{2} x+1}} d x\) Text Transcription: int cos x / sqrt{sin^2 x +1} dx

In Exercises 17-38, use integration tables to find the indefinite integral. \(\int \frac{e^{3 x}}{\left(1+e^{x}\right)^{3}} d x\) Text Transcription: int e^3x / (1 + e^x)^3 dx

In Exercises 17-38, use integration tables to find the indefinite integral. \(\int \cot ^{4} \theta d \theta\) Text Transcription: int cot^4 theta d theta

In Exercises 39–46, use integration tables to evaluate the definite integral. \(\int_{0}^{1} x e^{x^{2}} d x\) Text Transcription: int_{0}^{1} xe^{x^2} dx

In Exercises 39–46, use integration tables to evaluate the definite integral. \(\int_{0}^{4} \frac{x}{\sqrt{3+2 x}} d x\) Text Transcription: int_{0}^{4} x / sqrt{3 + 2x} dx

In Exercises 39–46, use integration tables to evaluate the definite integral. \(\int_{1}^{2} x^{4} \ln x d x\) Text Transcription: int_{1}^{2} x^4 ln x dx

In Exercises 39–46, use integration tables to evaluate the definite integral. \(\int_{0}^{\pi / 2} x \sin 2 x d x\) Text Transcription: int_{0}^{pi/2} x sin 2x dx

In Exercises 39–46, use integration tables to evaluate the definite integral. \(\int_{-\pi / 2}^{\pi / 2} \frac{\cos x}{1+\sin ^{2} x} d x\) Text Transcription: int_{-pi/2}^{pi/2} cos x / 1 + sin^2 x d x

In Exercises 39–46, use integration tables to evaluate the definite integral. \(\int_{0}^{5} \frac{x^{2}}{(5+2 x)^{2}} d x\) Text Transcription: int_{0}^{5} x^2 / (5 + 2x)^2 dx

In Exercises 39–46, use integration tables to evaluate the definite integral. \(\int_{0}^{\pi / 2} t^{3} \cos t d t\) Text Transcription: int_{0}^{pi/2} t^3 cos t dt

In Exercises 39–46, use integration tables to evaluate the definite integral. \(\int_{0}^{3} \sqrt{x^{2}+16} d x\) Text Transcription: \(\int_{0}^{3} \sqrt{x^{2}+16} d x\)

In Exercises 47-52, verify the integration formula. \(\int \frac{u^{2}}{(a+b u)^{2}} d u=\frac{1}{b^{3}}\left(b u-\frac{a^{2}}{a+b u}-2 a \ln |a+b u|\right)+C\) Text Transcription: int u^2 / (a + bu)^2 du = 1 / b^3 (bu - a^2 / a + bu} - 2a ln |a + bu|) + C

In Exercises 47-52, verify the integration formula. \(\int \frac{u^{n}}{\sqrt{a+b u}} d u=\frac{2}{(2 n+1) b}\left(u^{n} \sqrt{a+b u}-n a \int \frac{u^{n-1}}{\sqrt{a+b u}} d u\right)\) Text Transcription: int u^n / sqrt{a + bu} du = 2 / (2n + 1) b (u^n sqrt{a + bu} - na int u^{n - 1 / sqrt{a + bu}} du)

In Exercises 47-52, verify the integration formula. \(\int \frac{1}{\left(u^{2} \pm a^{2}\right)^{3 / 2}} d u=\frac{\pm u}{a^{2} \sqrt{u^{2} \pm a^{2}}}+C\) Text Transcription: int 1 / (u^{2} pm a^2)^3/2 du = pm u / a^2 sqrt{u^2 pm a^2} + C

In Exercises 47-52, verify the integration formula. \(\int u^{n} \cos u d u=u^{n} \sin u-n \int u^{n-1} \sin u d u\) Text Transcription: int u^{n} cos u du = u^n sin u - n int u^{n - 1} sin u du

In Exercises 47-52, verify the integration formula. \(\int \arctan u d u=u \arctan u-\ln \sqrt{1+u^{2}}+C\) Text Transcription: int arctan u du = u arctan u - ln sqrt{1 + u^2} + C

In Exercises 47-52, verify the integration formula. \(\int(\ln u)^{n} d u=u(\ln u)^{n}-n \int(\ln u)^{n-1} d u\) Text Transcription: int(ln u)^{n} du = u(ln u)^{n} - n int(ln u)^{n - 1} du

In Exercises 53-60, find or evaluate the integral. \(\int \frac{1}{2-3 \sin \theta} d \theta\) Text Transcription: int 1 / 2 - 3 sin theta d theta

In Exercises 53-60, find or evaluate the integral. \(\int \frac{\sin \theta}{1+\cos ^{2} \theta} d \theta\) Text Transcription: int sin theta / 1 + cos^2 theta d theta

In Exercises 53-60, find or evaluate the integral. \(\int_{0}^{\pi / 2} \frac{1}{1+\sin \theta+\cos \theta} d \theta\) Text Transcription: int_{0}^{pi/2} 1 / 1 + sin theta + cos theta d theta

In Exercises 53-60, find or evaluate the integral. \(\int_{0}^{\pi / 2} \frac{1}{3-2 \cos \theta} d \theta\) Text Transcription: int_{0}^{pi/2} 1 / 3 - 2 cos theta d theta

In Exercises 53-60, find or evaluate the integral. \(\int \frac{\sin \theta}{3-2 \cos \theta} d \theta\) Text Transcription: int sin theta / 3 - 2 cos theta d theta

In Exercises 53-60, find or evaluate the integral. \(\int \frac{\cos \theta}{1+\cos \theta} d \theta\) Text Transcription: int cos theta / 1 + cos theta d theta

In Exercises 53-60, find or evaluate the integral. \(\int \frac{\sin \sqrt{\theta}}{\sqrt{\theta}} d \theta\) Text Transcription: int sin sqrt theta / sqrt theta d theta

In Exercises 53-60, find or evaluate the integral. \(\int \frac{4}{\csc \theta-\cot \theta} d \theta\) Text Transcription: int 4 / csc theta - cot theta d theta

In Exercises 61 and 62, find the area of the region bounded by the graphs of the equations. \(y=\frac{x}{\sqrt{x+3}}, y=0, x=6\) Text Transcription: y = x / sqrt{x + 3}, y = 0, x = 6

In Exercises 61 and 62, find the area of the region bounded by the graphs of the equations. \(y=\frac{x}{1+e^{x^{2}}}, y=0, x=2\) Text Transcription: y = x / 1 + e^{x^2}, y = 0, x = 2

Finding a Pattern (a) Evaluate \(\int x^{n} \ln x d x\) for n = 1, 2, and 3. Describe any patterns you notice. (b) Write a general rule for evaluating the integral in part (a), for an integer \(n \geq 1\). Text Transcription: int x^{n} ln x dx n geq 1

Reduction Formula Describe what is meant by a reduction formula. Give an example.

State (if possible) the method or integration formula you would use to find the antiderivative. Explain why you chose that method or formula. Do not integrate. (a) \(\int \frac{e^{x}}{e^{2 x}+1} d x\) (b) \(\int \frac{e^{x}}{e^{x}+1} d x\) (c) \(\int x e^{x^{2}} d x\) (d) \(\int x e^{x} d x\) (e) \(\int e^{x^{2}} d x\) (f) \(\int e^{2 x} \sqrt{e^{2 x}+1} d x\) Text Transcription: int e^x / e^2x + 1 dx int e^x / e^x + 1 dx int xe^{x^2} dx int xe^{x} dx int e^{x^2} dx int e^{2x} sqrt{e^2x + 1} dx

Use the graph of f ‘ shown in the figure to answer the following. (a) Approximate the slope of \(f\) at x = - 1. Explain. (b) Approximate any open intervals in which the graph of \(f\) is increasing and any open intervals in which it is decreasing. Explain. Text Transcription: f

To use a table of integrals, the integral you are evaluating must appear in the table.

When using a table of integrals, you may have to make substitutions to rewrite your integral in the form in which it appears in the table.

Work A hydraulic cylinder on an industrial machine pushes a steel block a distance of \(x\) feet \((0 \leq x \leq 5)\), where the variable force required is \(F(x)=2000 x e^{-x}\) pounds. Find the work done in pushing the block the full 5 feet through the machine. Text Transcription: x (0 leq x leq 5) F(x) = 2000 xe^-x

Repeat Exercise 69, using \(F(x)=\frac{500 x}{\sqrt{26-x^{2}}}\) pounds. Text Transcription: F(x) = 500 x / sqrt{26 - x^2}

Consider the region bounded by the graphs of \(y=x \sqrt{16-x^{2}}, \quad y=0, \quad x=0,\) and x = 4. Find the volume of the solid generated by revolving the region about the \(y\)-axis. Text Transcription: y = x sqrt{16 - x^2}, y = 0, x = 0, y

The cross section of a precast concrete beam for a building is bounded by the graphs of the equations \(x=\frac{2}{\sqrt{1+y^{2}}}, \quad x=\frac{-2}{\sqrt{1+y^{2}}}, \quad y=0,\) and y = 3 where \(x\) and \(y\) are measured in feet. The length of the beam is 20 feet (see figure). (a) Find the volume V and the weight W of the beam. Assume the concrete weighs 148 pounds per cubic foot. (b) Find the centroid of a cross-section of the beam. Text Transcription: x = 2 / sqrt{1 + y^2}, x = -2 / sqrt{1 + y^2}, y = 0 x y

A population is growing according to the logistic model \(N=\frac{5000}{1+e^{4.8-1.9 t}}\) where \(t\) is the time in days. Find the average population over the interval [0,2]. Text Transcription: N = 5000 / 1 + e^{4.8 - 1.9 t} t

Evaluate \(\int_{0}^{\pi / 2} \frac{d x}{1+(\tan x)^{\sqrt{2}}}\). Text Transcription: int_{0}^{pi / 2} dx / 1 + (tan x)^{sqrt 2 }