?If \(f\) is continuous on \(\mathbb{R}\), prove that \(\int_{a}^{b} f(x+c) \ d

Evaluate the integral by making the given substitution. \(\int \cos 2 x\ d x, \quad u=2 x\) ________________ Equation Transcription: Text Transcription: integral cos? 2x dx, u=2x

Evaluate the integral by making the given substitution. \(\int x e^{-x^{2}} d x, u=-x^{2}\) Equation Transcription: Text Transcription: Integral xe^-x^2 dx, u=-x^2

Evaluate the integral by making the given substitution. \(\int x^{2} \sqrt{x^{3}+1} d x, u-x^{3}+1\) Equation Transcription: Text Transcription: integral x^2 square root x^3+1 dx, u=x^3+1

Evaluate the integral by making the given substitution. \(\int \sin ^{2} \theta \cos \theta d \theta, \quad u=\sin \theta\) Equation Transcription: Text Transcription: integral sin^2 theta cos theta d theta , u=sin theta

Evaluate the integral by making the given substitution. \(\int \frac{x^{3}}{x^{4}-5} d x, \quad u=x^{4}-5\) Equation Transcription: Text Transcription: Integral x^3/x^4-5dx, u=x^4-5

Evaluate the integral by making the given substitution. \(\int \frac{1}{x^{2}} \sqrt{1+\frac{1}{x}} d x, u=1+\frac{1}{x}\)

Evaluate the integral by making the given substitution. \(\int \frac{\cos \sqrt{t}}{\sqrt{t}} d t, \quad u=\sqrt{t}\) Equation Transcription: Text Transcription: Integral cos square root ?t/square root t dt, u=square root t

Evaluate the integral by making the given substitution. \(\int z \sqrt{z-1} d z, \quad u=z-1\) Equation Transcription: Text Transcription: Integral z square root z-1dz, u=z-1

Evaluate the indefinite integral. \(\int x \sqrt{1-x^{2}} d x\) Equation Transcription: Text Transcription: integral x square root 1-x^2 dx

Evaluate the indefinite integral. \(\int(5-3 x)^{10} d x\) ________________ Equation Transcription: Text Transcription: integral (5-3x)^10 dx

Evaluate the indefinite integral. \(\int t^{3} e^{-t^{4}} d t\) ________________ Equation Transcription: Text Transcription: integral t^3 e^-t^4 dt

Evaluate the indefinite integral. \(\int \sin t \sqrt{1+\cos t} d t\) ________________ Equation Transcription: Text Transcription: Integral sin ?t square root 1+cost dt

Evaluate the indefinite integral. \(\int \sin (\pi t / 3) d t\) Equation Transcription: Text Transcription: Integral sin(pi t/3)dt

Evaluate the indefinite integral. \(\int \sec ^{2} 2 \theta d \theta\) ________________ Equation Transcription: Text Transcription: Integral sec^2 ?2 theta d theta

Evaluate the indefinite integral. \(\int \frac{d x}{4 x+7}\)

Evaluate the indefinite integral. \(\int y^{2}\left(4-y^{3}\right)^{2 / 3} d y\) ________________ Equation Transcription: Text Transcription: Integral (4-y^3)^2/3 dy

Evaluate the indefinite integral. \(\int \frac{\cos \theta}{1+\sin \theta} d \theta\) Equation Transcription: Text Transcription: Integral cos theta/1+sin theta d theta

Evaluate the indefinite integral. \(\int \frac{z^{2}}{z^{3}+1} d z\) ________________ Equation Transcription: Text Transcription: integral z^2/z^3+1 dz

Evaluate the indefinite integral. \(\int \cos ^{3} \theta \sin \theta d \theta\) ________________ Equation Transcription: Text Transcription: Integral cos^3? theta sin? theta d theta

Evaluate the indefinite integral. \(\int e^{-5 r} d r\) ________________ Equation Transcription: Text Transcription: Integral e^-5r dr

Evaluate the indefinite integral. \(\int \frac{e^{u}}{\left(1-e^{u}\right)^{2}} d u\) Equation Transcription: Text Transcription: Integral e^u/(1-e^u)^2 du

Evaluate the indefinite integral. \(\int \frac{\sin (1 / x)}{x^{2}} d x\)

Evaluate the indefinite integral. \(\int \frac{a+b x^{2}}{\sqrt{3 a x+b x^{3}}} d x\)

Evaluate the indefinite integral. \(\int \frac{t+1}{3 t^{2}+6 t-5} d t\) ________________ Equation Transcription: Text Transcription: Integral t+1/3t^2+6t-5dt

Evaluate the indefinite integral. \(\int \frac{(\ln x)^{2}}{x} d x\) ________________ Equation Transcription: Text Transcription: Integral (ln? x)^2/x dx

Evaluate the indefinite integral. \(\int \sin x \sin (\cos x) d x\) Equation Transcription: Text Transcription: Integral sin x sin(cos x)dx

Evaluate the indefinite integral. \(\int \sec ^{2} \theta \tan ^{3} \theta d \theta\) ________________ Equation Transcription: Text Transcription: Integral sec^2 theta tan^3 theta d theta

Evaluate the indefinite integral. \(\int x \sqrt{x+2} d x\) Equation Transcription: Text Transcription: Integral x square root x+2 dx

Evaluate the indefinite integral. \(\int\left(x-\frac{1}{x^{2}}\right)\left(x^{2}+\frac{2}{x}\right)^{5} d x\) Equation Transcription: Text Transcription: Integral (x-1/x^2)(x^2+2/x)^5 dx

Evaluate the indefinite integral. \(\int \frac{d x}{a x+b}(a \neq 0)\) ________________ Equation Transcription: Text Transcription: Integral dx/ax+b(a not = 0)

Evaluate the indefinite integral. \(\int e^{r}\left(2+3 e^{r}\right)^{3 / 2} d r\) ________________ Equation Transcription: Text Transcription: Integral e^r(2+3e^r)^3/2 dr

Evaluate the indefinite integral. \(\int \frac{e^{\arcsin x}}{\sqrt{1-x^{2}}} d x\) Equation Transcription: Text Transcription: Integral e^arcsin x/square root 1-x^2 dx

Evaluate the indefinite integral. \(\int \frac{\sec ^{2} \theta}{\tan \theta} d \theta\)

Evaluate the indefinite integral. \(\int \frac{\sec ^{2} x}{\tan ^{2} x} d x\) ________________ Equation Transcription: Text Transcription: Integral sec^2 ?x/tan^2? x dx

Evaluate the indefinite integral. \(\int \frac{(\arctan x)^{2}}{x^{2}+1} d x\) Equation Transcription: Text Transcription: Integral (arctan x)^2/x^2+1dx

Evaluate the indefinite integral. \(\int \frac{1}{\left(x^{2}+1\right) \arctan x} d x\) Equation Transcription: Text Transcription: Integral 1/(x^2+1)arctan x dx

Evaluate the indefinite integral. \(\int 5^{t} \sin \left(5^{t}\right) d t\)

Evaluate the indefinite integral. \(\int \frac{\sin \theta \cos \theta}{1+\sin ^{2} \theta} d \theta\) Equation Transcription: Text Transcription: Integral sin theta cos theta/1 + sin^2 theta? d theta

Evaluate the indefinite integral. \(\int \cos (1+5 t) d t\)

Evaluate the indefinite integral. \(\int \frac{\cos (\pi / x)}{x^{2}} d x\) Equation Transcription: Text Transcription: Integral cos?(pi/x)/x^2 dx

Evaluate the indefinite integral. \(\int \sqrt{\cot x} \csc ^{2} x d x\) Equation Transcription: Text Transcription: Integral square root cot x csc^2 x dx

Evaluate the indefinite integral. \(\int \frac{2^{t}}{2^{t}+3} d t\) ________________ Equation Transcription: Text Transcription: Integral 2'/2'+3 dt

Evaluate the indefinite integral. \(\int \sinh ^{2} x \cosh x \ d x\) Equation Transcription: Text Transcription: Integral sinh^2 x cosh x dx

Evaluate the indefinite integral. \(\int \frac{d t}{\cos ^{2} t \sqrt{1+\tan t}}\)

Evaluate the indefinite integral. \(\int \frac{\sin 2 x}{1+\cos ^{2} x} \ d x\) Equation Transcription: Text Transcription: Integral sin 2x/1+cos^2 x dx

Evaluate the indefinite integral. \(\int \frac{\sin x}{1+\cos ^{2} x} \ d x\) Equation Transcription: Text Transcription: Integral sin 2x/1+cos^2 x dx

Evaluate the indefinite integral. \(\int \cot x \ d x\) Equation Transcription: Text Transcription: Integral cot x dx

Evaluate the indefinite integral. \(\int \frac{\cos (\ln t)}{t} \ d t\) Equation Transcription: Text Transcription: Integral cos (ln t)tdt

Evaluate the indefinite integral. \(\int \frac{d x}{\sqrt{1-x^{2}} \sin ^{-1} x}\) Equation Transcription: Text Transcription: Integral dx/sqrt 1-x^2 sin^-1 x

Evaluate the indefinite integral. \(\int \frac{x}{1+x^{4}} \ d x\) Equation Transcription: Text Transcription: Integral x/1+x^4 dx

Evaluate the indefinite integral. \(\int \frac{1+x}{1+x^{2}} \ d x\) Equation Transcription: Text Transcription: Integral 1+x/1+x^2 dx

Evaluate the indefinite integral. \(\int x^{2} \sqrt{2+x} \ d x\) Equation Transcription: Text Transcription: Integral x^2 sqrt 2+x dx

Evaluate the indefinite integral. \(\int x(2 x+5)^{8} \ d x\) Equation Transcription: Text Transcription: Integral x(2x+5)^8 dx

Evaluate the indefinite integral. \(\int x^{3} \sqrt{x^{2}+1} \ d x\) Equation Transcription: Text Transcription: Integral x^3 sqrt x^2+1 dx

Evaluate the indefinite integral. Illustrate and check that your answer is reasonable by graphing both the function and its antiderivative (take \(C=0\) ). \(\int x\left(x^{2}-1\right)^{3} \ d x\) Equation Transcription: Text Transcription: C=0 Integral x(x^2-1)^3 dx

Evaluate the indefinite integral. Illustrate and check that your answer is reasonable by graphing both the function and its antiderivative (take \(C=0\) ). \(\int \tan ^{2} \theta \sec ^{2} \theta \ d \theta\) Equation Transcription: Text Transcription: C=0 Integral tan^2theta sec^2theta dtheta

Evaluate the indefinite integral. Illustrate and check that your answer is reasonable by graphing both the function and its antiderivative (take \(C=0\) ). \(\int e^{\cos x} \sin x \ d x\) Equation Transcription: Text Transcription: C=0 Integral e^cos x sin x dx

Evaluate the indefinite integral. Illustrate and check that your answer is reasonable by graphing both the function and its antiderivative (take \(C=0\) ). \(\int \sin x \cos ^{4} x \ d x\) Equation Transcription: Text Transcription: C=0 Integral sin x cos^4 x dx

Evaluate the definite integral. \(\int_{0}^{1} \cos (\pi t / 2) \ d t\) Equation Transcription: Text Transcription: Integral over 0 ^1 cos (pit/2)dt

Evaluate the definite integral. \(\int_{0}^{1}(3 t-1)^{50} \ d t\) Equation Transcription: Text Transcription: Integral over 0 ^1 (3t-1)^50 dt

Evaluate the definite integral. \(\int_{0}^{1} \sqrt[3]{1+7 x} \ d x\) Equation Transcription: Text Transcription: Integral over 0 ^1 sqrt ^3_1+7x dx

Evaluate the definite integral. \(\int_{\pi / 3}^{2 \pi / 3} \csc ^2\left(\frac{1}{2} t\right) d t\)

Evaluate the definite integral. \(\int_{0}^{\pi / 6} \frac{\sin t}{\cos ^{2} t} \ d t\) Equation Transcription: Text Transcription: Integral over 0 pi/6 sin t/cos^2 t dt

Evaluate the definite integral. \(\int_{1}^{4} \frac{\sqrt{2+\sqrt{x}}}{\sqrt{x}} \ d x\) Equation Transcription: Text Transcription: Integral over 1 ^4 2+sqrt x/sqrt x dx

Evaluate the definite integral. \(\int_{1}^{2} \frac{e^{1 / x}}{x^{2}} \ d x\)

Evaluate the definite integral. \(\int_{0}^{1} \frac{e^{x}}{1+e^{2 x}} \ d x\) Equation Transcription: Text Transcription: Integral over 0^1 e^x/1+e^2x dx

Evaluate the definite integral. \(\int_{-\pi / 4}^{\pi / 4}\left(x^{3}+x^{4} \tan x\right) \ d x\) Equation Transcription: Text Transcription: Integral over -pi/4 ^pi/4(x^3+x^4 tan x)dx

Evaluate the definite integral. \(\int_{0}^{\pi / 2} \cos x \sin (\sin x) \ d x\) Equation Transcription: Text Transcription: Integral over 0^pi/2cos x sin(sin x)dx

Evaluate the definite integral. \(\int_{0}^{13} \frac{d x}{\sqrt[3]{(1+2 x)^{2}}}\) Equation Transcription: Text Transcription: Integral over 0^13 dx/sqrt^3_(1+2x)^2

Evaluate the definite integral. \(\int_{0}^{a} x \sqrt{a^{2}-x^{2}} \ d x\) Equation Transcription: Text Transcription: Integral over 0^a x sqrt a^2-x^2 dx

Evaluate the definite integral. \(\int_{0}^{a} x \sqrt{x^{2}+a^{2}} \ d x \quad(a>0)\) Equation Transcription: Text Transcription: Integral over 0 ^a x sqrt x^2+a^2 dx (a>0)

Evaluate the definite integral. \(\int_{-\pi / 3}^{\pi / 3} x^{4} \sin x \ d x\) Equation Transcription: Text Transcription: Integral over -pi/3 ^pi/3 x^4 sin x dx

Evaluate the definite integral. \(\int_{1}^{2} x \sqrt{x-1} \ d x\) Equation Transcription: Text Transcription: Integral over 1 ^2 x sqrt x-1 dx

Evaluate the definite integral. \(\int_{0}^{4} \frac{x}{\sqrt{1+2 x}} \ d x\) Equation Transcription: Text Transcription: Integral over 0 ^4 x/sqrt 1+2x dx

Evaluate the definite integral. \(\int_{e}^{e^{4}} \frac{d x}{x \sqrt{\ln x}}\)

Evaluate the definite integral. \(\int_{0}^{2}(x-1) e^{(x-1)^{2}} \ d x\) Equation Transcription: Text Transcription: Integral over 0^2 (x-1)e^(x-1)^2 dx

Evaluate the definite integral. \(\int_{0}^{1} \frac{e^{z}+1}{e^{z}+z} \ d z\) Equation Transcription: Text Transcription: Integral over 0 ^1 e^z+1/e^z+z dz

Evaluate the definite integral. \(\int_{1}^{4} \frac{1}{(x+1) \sqrt{x}} \ d x\) Equation Transcription: Text Transcription: Integral over 1 ^4 1/(x+1)sqrt x dx

Evaluate the definite integral. \(\int_{0}^{1} \frac{d x}{(1+\sqrt{x})^{4}}\) Equation Transcription: Text Transcription: Integral over 0^1 dx/(1+sqrt x)^4

Evaluate the definite integral. \(\int_{1}^{16} \frac{x^{1 / 2}}{1+x^{3 / 4}} \ d x\) Equation Transcription: Text Transcription: Integral over 1^16 x^1/2 1/1+x^3/4 dx

Use a graph to give a rough estimate of the area of the region that lies under the given curve. Then find the exact area. \(y=\sqrt{2 x+1}, 0 \leqslant x \leqslant 1\) Equation Transcription: ?? Text Transcription: y=sqrt 2x+1, 0 leqsslant x leqslant 1

Use a graph to give a rough estimate of the area of the region that lies under the given curve. Then find the exact area. \(y=2 \sin x-\sin 2 x, \quad 0 \leqslant x \leqslant \pi\)

Evaluate \(\int_{-2}^{2}(x+3) \sqrt{4-x^{2}} \ d x\) by writing it as a sum of two integrals and interpreting one of those integrals in terms of an area.

Evaluate \(\int_{0}^{1} x \sqrt{1-x^{4}} \ d x\) by making a substitution and interpreting the resulting integral in terms of an area. Equation Transcription: Text Transcription: Integral over 0 ^1 x sqrt 1-x^4 dx

Which of the following areas are equal? Why?

A model for the basal metabolism rate, in \(kcal/h\), of a young man is \(R(t)=85-0.18 \cos (\pi t / 12)\), where \(t\) is the time in hours measured from 5:00 AM. What is the total basal metabolism of this man, \(\int_{0}^{24} R(t) d t\), over a 24 -hour time period? Equation Transcription: Text Transcription: kcal/h R(t)=85-0.18 cos(pi/12) t Integral over 0^24 R(t) dt

An oil storage tank ruptures at time \(t=0\) and oil leaks from the tank at a rate of \(r(t)=100 e^{-0.01 t}\) liters per minute. How much oil leaks out during the first hour? Equation Transcription: Text Transcription: t=0 r(t)=100^-0.01t

A bacteria population starts with 400 bacteria and grows at a rate of \(r(t)=(450.268) e^{1.12567 t}\) bacteria per hour. How many bacteria will there be after three hours? Equation Transcription: Text Transcription: r(t)=(450.268)e^1.12567t

Breathing is cyclic and a full respiratory cycle from the beginning of inhalation to the end of exhalation takes about 5 seconds. The maximum rate of air flow into the lungs is about \(0.5 \mathrm{~L} / \mathrm{s}\). This explains, in part, why the function \(f(t)=\frac{1}{2} \sin (2 \pi t / 5)\) has often been used to model the rate of air flow into the lungs. Use this model to find the volume of inhaled air in the lungs at time $t$. Equation Transcription: Text Transcription: 0.5 L/s f(t)=1/2 sin (2pi t/5 t

The rate of growth of a fish population was modeled by the equation \(G(t)=\frac{60,000 e^{-0.6 t}}{\left(1+5 e^{-0.6 t}\right)^{2}}\) where \(t\) is the number of years since 2000 and \(G\) is measured in kilograms per year. If the biomass was \(25,000 \mathrm{~kg}\) in the year 2000 , what is the predicted biomass for the year 2020? Equation Transcription: Text Transcription: G(t)=60,000^e-0.6t/(1+5e^-0.6t)^2 t G 25,000 kg

Dialysis treatment removes urea and other waste products from a patient's blood by diverting some of the blood flow externally through a machine called a dialyzer. The rate at which urea is removed from the blood (in mg/min) is often well described by the equation \(u(t)=\frac{r}{V} C_{0} e^{-r t / V}\) where \(r\) is the rate of flow of blood through the dialyzer(in mL/min), \(V\) is the volume of the patient's blood (in mL), and \(C_{0}\) is the amount of urea in the blood (in mg) at time \(t=0\). Evaluate the integral \(\int_{0}^{30} u(t) d t\) and interpret it. Equation Transcription: Text Transcription: u(t)=r/V C_0 e^-rt/V r V C0 t=0 Integral over 0 ^30 u(t)dt

Alabama Instruments Company has set up a production line to manufacture a new calculator. The rate of production of these calculators after t weeks is \(\frac{d x}{d t}=5000\left(1-\frac{100}{(t+10)^{2}}\right)\) calculators/week (Notice that production approaches 5000 per week as time goes on, but the initial production is lower because of the workers' unfamiliarity with the new techniques.) Find the number of calculators produced from the beginning of the third week to the end of the fourth week.

If \(f\) is continuous and \(\int_{0}^{4} f(x) d x=10\), find \(\int_{0}^{2} f(2 x) d x\). Equation Transcription: Text Transcription: f Integral over 0^4 f(x)dx=10 Integral over 0^2 f(2x)dx

If \(f\) is continuous and \(\int_{0}^{9} f(x) d x=4\), find \(\int_{0}^{3} x f\left(x^{2}\right) d x\). Equation Transcription: Text Transcription: f Integral over 0 ^9 f(x)dx=4 Integral over 0 ^3 xf(x^2)dx

If f is continuous on \(\mathbb{R}\), prove that \(\int_{a}^{b} f(-x) d x=\int_{-b}^{-a} f(x) d x\) For the case where \(f(x) \geqslant 0\) and 0 < a < b, draw a diagram to interpret this equation geometrically as an equality of areas.

If \(f\) is continuous on \(\mathbb{R}\), prove that \(\int_{a}^{b} f(x+c) \ d x=\int_{a+c}^{b+c} f(x) \ d x\) For the case where \(f(x) \geqslant 0\), draw a diagram to interpret this equation geometrically as an equality of areas. Equation Transcription: ? ? Text Transcription: f R Integral over a ^b f(x+c)dx=integral over a+c ^b+cf(x)dx f(x)geqslant 0

If \(a\) and \(b\) are positive numbers, show that \(\int_{0}^{1} x^{a}(1-x)^{b} \ d x=\int_{0}^{1} x^{b}(1-x)^{a} \ d x\) Equation Transcription: Text Transcription: a b Integral over 0 ^1 x^a (1-x)^b dx=Integral over 0 ^1 x^b(1-x)^a dx

If \(f\) is continuous on \([0, \pi]\), use the substitution \(\bar{u}=\pi-x\) to show that \(\int_{0}^{\pi} x f(\sin x) d x=\frac{\pi}{2} \int_{0}^{\pi} f(\sin x) \ d x\) Equation Transcription: Text Transcription: f [0,pi] u=pi-x Integral over 0 ^pi xf(sin x) dx=pi/2 integral over 0 ^pi f(sin x) dx

Use Exercise 98 to evaluate the integral \(\int_{0}^{\pi} \frac{x \sin x}{1+\cos ^{2} x} d x\)

(a) If f is continuous, prove that \(\int_{0}^{\pi / 2} f(\cos x) \ d x=\int_{0}^{\pi / 2} f(\sin x) \ d x\) (b) Use part (a) to evaluate \(\int_{0}^{\pi / 2} \cos ^{2} x \ d x\) and \(\int_{0}^{\pi / 2} \sin ^{2} x \ d x\)