?The Laplace Transform If is continuous for , the Laplace transform of is the function

Explain why each of the following integrals is improper. (a) \(\int_{1}^{4} \frac{d x}{x\ -\ 3}\) (b) \(\int_{3}^{\infty} \frac{d x}{x^{2}\ -\ 4}\) (c) \(\int_{0}^{1} \tan \pi x \ d x\) (d) \(\int_{-\infty}^{-1} \frac{e^{x}}{x} \ d x\) Equation Transcription: ? ? ? tan x dx ? dx Text Transcription: integral _1 ^4 dx/x-3 integral _3 ^infinity dx/x^2-4 integral _0 ^1 tan pi x dx integral _-infinity ^-1 e^x/x dx

Which of the following integrals are improper? Why? (a) \(\int_{0}^{\pi} \sec x \ d x\) (b) \(\int_{0}^{4} \frac{d x}{x\ -\ 5}\) (c) \(\int_{-1}^{3} \frac{d x}{x\ +\ x^{3}}\) (d) \(\int_{1}^{\infty} \frac{d x}{x\ +\ x^{3}}\)

Find the area under the curve \(y=1 / x^{3}\) from \(x=1\) to \(x=t\), and evaluate it for t = 10, 100, and 1000 . Then find the total area under this curve for \(x \geqslant 1\).

(a) Graph the functions \(f(x)=1 / x^{1.1}\) and \(g(x)=1 / x^{0.9}\) in both the viewing rectangles [0,10] by [0,1] and [0,100] by [0,1]. (b) Find the areas under the graphs of \(f\) and \(g\) from \(x=1\) to \(x=t\) and evaluate for \(t=10,100,10^{4}, 10^{6}, 10^{10}\), and \(10^{20}\). (c) Find the total area under each curve for \(x \geqslant 1\), if it exists. ________________ Equation Transcription: ? 1 Text Transcription: f(x)=1 / x^{1.1} g(x)=1 / x^{0.9} f g x = 1 x = t t = 10,100,10^{4}, 10^{6}, 10^{10} 10^{20} x geqslant 1

Determine whether the integral is convergent or divergent. Evaluate integrals that are convergent. \(\int_{1}^{\infty} 2 x^{-3} d x\) ________________ Equation Transcription: Text Transcription: int_{1}^{\infty} 2 x^{-3} dx

Determine whether the integral is convergent or divergent. Evaluate integrals that are convergent. \(\int_{-\infty}^{-1} \frac{1}{\sqrt[3]{x}} \ d x\)

Determine whether the integral is convergent or divergent. Evaluate integrals that are convergent. \(\int_{0}^{\infty} e^{-2 x} d x\) ________________ Equation Transcription: Text Transcription: int_{0}^{infty} e^{-2 x} dx

Determine whether the integral is convergent or divergent. Evaluate integrals that are convergent. \(\int_{1}^{\infty}\left(\frac{1}{3}\right)^{x} d x\) ________________ Equation Transcription: Text Transcription: int_{1}^{infty}(frac{1}{3})^{x} dx

Determine whether the integral is convergent or divergent. Evaluate integrals that are convergent. \(\int_{-2}^{\infty} \frac{1}{x+4} d x\) ________________ Equation Transcription: Text Transcription: int_{-2}^{infty} frac{1}{x+4} dx

Determine whether the integral is convergent or divergent. Evaluate integrals that are convergent. \(\int_{1}^{\infty} \frac{1}{x^{2}+4} d x\) Equation Transcription: Text Transcription: Integral 1 infinity 1/x^2+4 dx

Determine whether the integral is convergent or divergent. Evaluate integrals that are convergent.

Determine whether the integral is convergent or divergent. Evaluate integrals that are convergent.

Determine whether the integral is convergent or divergent. Evaluate integrals that are convergent. \(\int_{-\infty}^{0} \frac{x}{\left(x^{2}+1\right)^{3}} d x\) Equation Transcription: Text Transcription: Integral - infinity 0 x/(x^2+1) dx

Determine whether the integral is convergent or divergent. Evaluate integrals that are convergent. \(\int_{-\infty}^{-3} \frac{x}{4-x^{2}} d x\) Equation Transcription Text Transcription: Integral - infinity - 3 x/ 4-x^2 dx

Determine whether the integral is convergent or divergent. Evaluate integrals that are convergent. \(\int_{1}^{\infty} \frac{x^{2}+x+1}{x^{4}} d x\) Equation Transcription: Text Transcription: Integral 1 infinity x^2+x+1/x^4 dx

Determine whether the integral is convergent or divergent. Evaluate integrals that are convergent. \(\int_{2}^{\infty} \frac{x}{\sqrt{x^{2}-1}} d x\) Equation Transcription: Text Transcription: Integral 2 infinity x/ square root x^2 - 1 dx

Determine whether the integral is convergent or divergent. Evaluate integrals that are convergent. \(\int_{0}^{\infty} \frac{e^{x}}{\left(1+e^{x}\right)^{2}} d x\) Equation Transcription: Text Transcription: Integral 0 infinity e^x/(1+e^x)^2 dx

Determine whether the integral is convergent or divergent. Evaluate integrals that are convergent. \(\int_{-\infty}^{-1} \frac{x^{2}+x}{x^{3}} d x\) Equation Transcription: Text Transcription: Integral - infinity - 1 x^2+x/x^3 dx

Determine whether the integral is convergent or divergent. Evaluate integrals that are convergent. \(\int_{-\infty}^{\infty} x e^{-x^{2}} d x\) Equation Transcription: Text Transcription: Integral - infinity infinity xe^-x^2 dx

Determine whether the integral is convergent or divergent. Evaluate integrals that are convergent. \(\int_{-\infty}^{\infty} \frac{x}{x^{2}+1} d x\) Equation Transcription: Text Transcription: Integral - infinity infinity x/x^2+1 dx

Determine whether the integral is convergent or divergent. Evaluate integrals that are convergent. \(\int_{-\infty}^{\infty} \cos 2 t \ d t\) Equation Transcription: ? cos 2t dt Text Transcription: integral _-infinity ^infinity cos 2t dt

Determine whether the integral is convergent or divergent. Evaluate integrals that are convergent. \(\int_{1}^{\infty} \frac{e^{-1 / x}}{x^{2}} \ d x\)

Determine whether the integral is convergent or divergent. Evaluate integrals that are convergent. \(\int_{0}^{\infty} \sin ^{2} \alpha \ d \alpha\) Equation Transcription: ? sin2 d Text Transcription: integral _0 ^infinity sin^2 alpha d alpha

Determine whether the integral is convergent or divergent. Evaluate integrals that are convergent. \(\int_{0}^{\infty} \sin \theta \quad e^{\cos \theta} \quad d \theta\) Equation Transcription: ? sin ecos d Text Transcription: integral _0 ^infinity sin theta e^cos theta d theta

Determine whether the integral is convergent or divergent. Evaluate integrals that are convergent. \(\int_{1}^{\infty} \frac{1}{x^{2}+x} d x\) Equation Transcription: Text Transcription: Integral 1 infinity 1/x^2+x dx

Determine whether the integral is convergent or divergent. Evaluate integrals that are convergent. \(\int_{2}^{\infty} \frac{d v}{v^{2}+2 v-3}\) Equation Transcription: Text Transcription: Integral 2 infinity dv/v^2++2v-3

Determine whether the integral is convergent or divergent. Evaluate integrals that are convergent. \(\int_{-\infty}^{0} z e^{2 z} d z\) Equation Transcription: Text Transcription: Integral -infinity 0 ze^2z dz

Determine whether the integral is convergent or divergent. Evaluate integrals that are convergent.

Determine whether the integral is convergent or divergent. Evaluate integrals that are convergent. \(\int_{1}^{\infty} \frac{\ln x}{x} d x\) Equation Transcription: ? dx Text Transcription: integral _1 ^infinity ln x/x dx

Determine whether the integral is convergent or divergent. Evaluate integrals that are convergent. \(\int_{1}^{\infty} \frac{\ln x}{x^{2}} d x\) Equation Transcription: ? dx Text Transcription: integral _1 ^infinity ln x/x^2 dx

Determine whether the integral is convergent or divergent. Evaluate integrals that are convergent. \(\int_{-\infty}^{0} \frac{z}{z^{4}+4} \ d z\)

Determine whether the integral is convergent or divergent. Evaluate integrals that are convergent. \(\int_e^{\infty} \frac{1}{x(\ln x)^2} d x\)

Determine whether the integral is convergent or divergent. Evaluate integrals that are convergent. \(\int_{0}^{\infty} e^{-\sqrt{y}} d y\) ________________ Equation Transcription: Text Transcription: int_{0}^{infty} e^{-sqrt{y}} dy

Determine whether the integral is convergent or divergent. Evaluate integrals that are convergent. \(\int_{1}^{\infty} \frac{d x}{\sqrt{x}+x \sqrt{x}}\) ________________ Equation Transcription: Text Transcription: int_{1}^{infty} frac{d x}{sqrt{x} + x sqrt{x}}

Determine whether the integral is convergent or divergent. Evaluate integrals that are convergent. \(\int_{0}^{1} \frac{1}{x} d x\) Equation Transcription: Text Transcription: Integral 0 1 1/x dx

Determine whether the integral is convergent or divergent. Evaluate integrals that are convergent. \(\int_{0}^{5} \frac{1}{\sqrt[3]{5-x}} d x\) ________________ Equation Transcription: Text Transcription: int_{0}^{5} \frac{1}{\sqrt[3]{5-x}} dx

Determine whether the integral is convergent or divergent. Evaluate integrals that are convergent. \(\int_{-2}^{14} \frac{d x}{\sqrt[4]{x+2}}\) Equation Transcription: Text Transcription: Integral -2 14 dx/ 4 square root x+2

Determine whether the integral is convergent or divergent. Evaluate integrals that are convergent. \(\int_{-1}^{2} \frac{x}{(x+1)^{2}} d x\) Equation Transcription: Text Transcription: Integral -1 2 x/(x+1)^2 dx

Determine whether the integral is convergent or divergent. Evaluate integrals that are convergent. \(\int_{-2}^{3} \frac{1}{x^{4}} d x\) ________________ Equation Transcription: Text Transcription: int_{-2}^{3} frac{1}{x^{4}} dx

Determine whether the integral is convergent or divergent. Evaluate integrals that are convergent. \(\int_{0}^{1} \frac{d x}{\sqrt{1-x^{2}}}\) Equation Transcription: Text Transcription: Integral 0 1 dx/ square root 1 - x^2

Determine whether the integral is convergent or divergent. Evaluate integrals that are convergent. \(\int_0^9 \frac{1}{\sqrt[3]{x-1}} d x\)

Determine whether the integral is convergent or divergent. Evaluate integrals that are convergent. \(\int_{0}^{5} \frac{w}{w-2} d w\) ________________ Equation Transcription: Text Transcription: int_{0}^{5} frac{w}{w-2} dw

Determine whether the integral is convergent or divergent. Evaluate integrals that are convergent. \(\int_{0}^{\pi / 2} \tan ^{2} \theta \quad d \theta\) Equation Transcription: ? tan2 d Text Transcription: integral _0 ^pi/2 tan^2 theta d theta

Determine whether the integral is convergent or divergent. Evaluate integrals that are convergent. \(\int_{0}^{4} \frac{d x}{x^{2}-x-2}\) Equation Transcription: Text Transcription: Integral 0 4 dx/x^2-x-2

Determine whether the integral is convergent or divergent. Evaluate integrals that are convergent. \(\int_{0}^{1} r \ln r \quad d r\) Equation Transcription: ? r ln r dr Text Transcription: integral _0 ^1 r ln r dr

Determine whether the integral is convergent or divergent. Evaluate integrals that are convergent. \(\int_{0}^{\pi / 2} \frac{\cos \theta}{\sqrt{\sin \theta}} d \theta\) Equation Transcription: ?d Text Transcription: integral _0 ^pi/2 cos theta/sqrt sin theta d theta

Determine whether the integral is convergent or divergent. Evaluate integrals that are convergent. \(\int_{-1}^{0} \frac{e^{1 / x}}{x^{3}} \ d x\)

Determine whether the integral is convergent or divergent. Evaluate integrals that are convergent.

Sketch the region and find its area (if the area is finite). \(S=\left\{(x, \ y) \mid x \geqslant 1, \ 0 \leqslant y \leqslant e^{-x}\right\}\)

Sketch the region and find its area (if the area is finite). \(S=\left\{(x, \ y) \mid x \leqslant 0, \ 0 \leqslant y \leqslant e^{x}\right\}\)

Sketch the region and find its area (if the area is finite).

Sketch the region and find its area (if the area is finite).

Sketch the region and find its area (if the area is finite). \(S=\left\{(x, y) \mid 0 \leqslant x<\pi / 2,0 \leqslant y \leqslant \sec ^{2} x\right\}\) Equation Transcription: S = {(x, y) | 0 ? x < /2, 0 ? y ? sec2x} Text Transcription: S = {(x, y) mid 0 leqslant x < pi/2, 0 leqslant y leqslant sec^2x}

Sketch the region and find its area (if the area is finite).

(a) If \(g(x)=\left(\sin ^{2} x\right) / x^{2}\), use a calculator or computer to make a table of approximate values of \(\int_{1}^{t} g(x) \quad d x\) for \(t=2,5,10,100,1000, \text { and } 10,000\). Does it appear that \(\int_{1}^{\infty} g(x) \quad d x\) is convergent? (b) Use the Comparison Theorem with \(f(x)=1 / x^{2}\) to show that \(\int_{1}^{\infty} g(x) \quad d x\) is convergent. (c) Illustrate part (b) by graphing \(f\) and \(g\) on the same screen for \(1 \leqslant x \leqslant 10\). Use your graph to explain intuitively why \(\int_{1}^{\infty} g(x) \quad d x\) is convergent. Equation Transcription: g(x) = (sin2x)/x2 ?g(x) dx t = 2, 5, 10, 100, 1000, and 10,000 ?g(x) dx f(x) = 1/x2 ?g(x) dx f g 1 ? x ? 10 ?g(x) dx Text Transcription: g(x) = (sin^2x)/x^2 integral _1 ^t g(x) dx t = 2, 5, 10, 100, 1000, and 10,000 integral _1 ^infinity g(x) dx f(x) = 1/x^2 integral _1 ^infinity g(x) dx f g 1 leqslant x leqslant 10 integral _1 ^infinity g(x) dx

(a) If \(g(x)=1 /(\sqrt{x}-1\), use a calculator or computer to make a table of approximate values of \(\int_{2}^{t} g(x) d x\) for , and Does it appear that \(\int_{2}^{\infty} g(x) d x\) is convergent or divergent? (b) Use the Comparison Theorem with \(f(x)=1 / \sqrt{x}\) to show that \(\int_{2}^{\infty} g(x) d x\) is divergent. (c) Illustrate part (b) by graphing and on the same screen for \(2 \leqslant x \leqslant 20\) . Use your graph to explain intuitively why \(\int_{2}^{\infty} g(x) d x\) is divergent. Equation Transcription: ?? Text Transcription: g(x) = 1/(square root x-1) Integral 2 t g(x)dx Integral 2 infinity g(x)dx f(x)=1/square root x Integral 2 infinity g(x)dx 2 leqslant x leqslant 20 Integral 2 infinity g(x)dx

Use the Comparison Theorem to determine whether the integral is convergent or divergent.

Use the Comparison Theorem to determine whether the integral is convergent or divergent. \(\int_{1}^{\infty} \frac{1+\sin ^{2} x}{\sqrt{x}} d x\) Equation Transcription: ?dx Text Transcription: integral _1 ^infinity 1+sin^2x/sqrt x dx

Use the Comparison Theorem to determine whether the integral is convergent or divergent. \(\int_{2}^{\infty} \frac{1}{x-\ln x} d x\) Equation Transcription: ?dx Text Transcription: integral _2 ^infinity 1/x-ln x dx

Use the Comparison Theorem to determine whether the integral is convergent or divergent. \(\int_{0}^{\infty} \frac{\arctan x}{2+e^{x}} \ d x\)

Use the Comparison Theorem to determine whether the integral is convergent or divergent.

Use the Comparison Theorem to determine whether the integral is convergent or divergent. \(\int_{1}^{\infty} \frac{2+\cos x}{\sqrt{x^{4}+x^{2}}} d x\) Equation Transcription: Text Transcription: integral _1 ^infinity 2+cos x/sqrt x^4+x^2 dx

Use the Comparison Theorem to determine whether the integral is convergent or divergent. \(\int_{0}^{1} \frac{\sec ^{2} x}{x \sqrt{x}} d x\) Equation Transcription: Text Transcription: integral _0 ^1 sec^2 x/x sqrt x dx

Use the Comparison Theorem to determine whether the integral is convergent or divergent. \(\int_{0}^{\pi} \frac{\sin ^{2} x}{\sqrt{x}} d x\) Equation Transcription: Text Transcription: integral _0 ^pi sin^2 x/sqrt x dx

Improper Integrals that Are Both Type 1 and Type 2 The integral \(\int_{a}^{\infty} f(x) \ d x\) is improper because the interval \([a, \ \infty)\) is infinite. If f has an infinite discontinuity at a, then the integral is improper for a second reason. In this case we evaluate the integral by expressing it as a sum of improper integrals of Type 2 and Type 1 as follows: \(\int_{a}^{\infty} f(x) \ d x=\int_{a}^{c} f(x) \ d x+\int_{c}^{\infty} f(x) \ d x\) \(c>a\) Evaluate the given integral if it is convergent. \(\int_{0}^{\infty} \frac{1}{x^{2}} \ d x\)

Improper Integrals that Are Both Type 1 and Type 2 The integral is improper because the interval is infinite. If has an infinite discontinuity at , then the integral is improper for a second reason. In this case we evaluate the integral by expressing it as a sum of improper integrals of Type 2 and Type 1 as follows: Evaluate the given integral if it is convergent.

Improper Integrals that Are Both Type 1 and Type 2 The integral \(\int_{a}^{\infty} f(x) d x\) is improper because the interval \([a, \infty)\). If f has an infinite discontinuity at a, then the integral is improper for a second reason. In this case we evaluate the integral by expressing it as a sum of improper integrals of Type 2 and Type 1 as follows: \(\int_{a}^{\infty} f(x) d x=\int_{a}^{c} f(x) d x+\int_{c}^{\infty} f(x) d x \quad c>a\) Evaluate the given integral if it is convergent. \(\int_{0}^{\infty} \frac{1}{\sqrt{x}(1+x)} d x\)

Improper Integrals that Are Both Type 1 and Type 2 The integral \(\int_{a}^{\infty} f(x) d x\) is improper because the interval \([a, \infty)\) is infinite. If \(f\) has an infinite discontinuity at \(a\), then the integral is improper for a second reason. In this case we evaluate the integral by expressing it as a sum of improper integrals of Type 2 and Type 1 as follows: \(\int_{a}^{\infty} f(x) d x=\int_{a}^{c} f(x) d x+\int_{c}^{\infty} f(x) d x c>a\) Evaluate the given integral if it is convergent. \(\int_{2}^{\infty} \frac{1}{x \sqrt{x^{2}-4}} d x\) Equation Transcription: Text Transcription: integral_1^infinity f(x)dx [a,infinity) f a integral _1 ^infinity f(x) dx = integral_a^infinity f(x) dx = integral_c ^infinity f(x) c>a integral _1 ^infinity 2+cos x/sqrt x^4+x^2 dx

Find the values of \(p\) for which the integral converges and evaluate the integral for those values of \(p\). \(\int_{0}^{1} \frac{1}{x^{p}} \ d x\) Equation Transcription: Text Transcription: p p int_{0}^{1} \frac{1}{x^{p}} d x

Find the values of \({p}\) for which the integral converges and evaluate the integral for those values of \(\mu\). \(\int_{e}^{\infty} \frac{1}{x(\ln x)^{p}} d x\) Equation Transcription: p ?dx Text Transcription: p mu integral _e ^infinity 1/x(ln x)^p dx

Find the values of p for which the integral converges and evaluate the integral for those values of \(\mu\). \(\int_{0}^{1} x^{p} \ln x \quad d x\)

(a) Evaluate the integral for , and 3 . (b) Guess the value of when is an arbitrary positive integer. (c) Prove your guess using mathematical induction.

The Cauchy principal value of the integral is defined by Show that diverges but the Cauchy principal value of this integral is

The average speed of molecules in an ideal gas is \(\bar{v}=\frac{4}{\sqrt{\pi}}\left(\frac{M}{2 R T}\right)^{3 / 2} \int_{0}^{\infty} v^{3} e^{\left.-M v^{2} / 2 R T\right)} d v\) where is the molecular weight of the gas, is the gas constant, is the gas temperature, and is the molecular speed. Show that \(\bar{v}=\sqrt{\frac{8 R T}{\pi M}}\) Equation Transcription: Text Transcription: v bar = 3/square root of pi (M/2RT)^3/2 integral from 0 to infinity v^3 e^-Mv^2/2RT) dv v bar = square root 8RT/piM

We know from Example 1 that the region has infinite area. Show that by rotating about the -axis we obtain a solid (called Gabriel's horn) with finite volume.

Use the information and data in Exercise to find the work required to propel a space vehicle out of the earth's gravitational field.

Find the escape velocity that is needed to propel a rocket of mass out of the gravitational field of a planet with mass and radius . Use Newton's Law of Gravitation (see Exercise 6.4.35) and the fact that the initial kinetic energy of supplies the needed work.

Astronomers use a technique called stellar stereography to determine the density of stars in a star cluster from the observed (two-dimensional) density that can be analyzed from a photograph. Suppose that in a spherical cluster of radius \(R\) the density of stars depends only on the distance \(r\) from the center of the cluster. If the perceived star density is given by \(y(s)\), where \(s\) is the observed planar distance from the center of the cluster, and \(x(r)\) is the actual density, it can be shown that \(y(s)=\int_{s}^{R} \frac{2 r}{\sqrt{r^{2}-s^{2}}} x(r) d r\) If the actual density of stars in a cluster is \(x(r)=\frac{1}{2}(R-r)^{2}\) find the perceived density \(y(s)\). Equation Transcription: Text Transcription: R r y(s) s x(r) y(s)= integral s ^R 2r/ square root r^2-s^2 x (r) dr x(r) =½ (R-r)^2 y(s)

A manufacturer of lightbulbs wants to produce bulbs that last about 700 hours but, of course, some bulbs burn out faster than others. Let be the fraction of the company's bulbs that burn out before hours, so always lies between 0 and 1 . (a) Make a rough sketch of what you think the graph of might look like. (b) What is the meaning of the derivative (c) What is the value of ? Why?

As we saw in Section , a radioactive substance decays exponentially: The mass at time is , where is the initial mass and is a negative constant. The mean life of an atom in the substance is For the radioactive carbon isotope, , used in radiocarbon dating, the value of is . Find the mean life of a atom.

In a study of the spread of illicit drug use from an enthusiastic user to a population of users, the authors model the number of expected new users by the equation where , and are positive constants. Evaluate this integral to express in terms of , and .

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 ) is often well described by the equation where is the rate of flow of blood through the dialyzer (in ), is the volume of the patient's blood (in ), and is the amount of urea in the blood (in ) at time Evaluate the integral and interpret it.

Determine how large the number a has to be so that \(\int_{a}^{\infty} \frac{1}{x^{2}+1} d x<0.001\) Text Transcription: int_{a}^{infty} 1 / x^2 + 1 dx < 0.001

Estimate the numerical value of \(\int_{0}^{\infty} e^{-x^{2}} d x\) by writing it as the sum of \(\int_{0}^{4} e^{-x^{2}} d x\)and \(\int_{4}^{\infty} e^{-x^{2}} d x\). Approximate the first integral by using Simpson's Rule with \(n=8\) and show that the second integral is smaller than\(\int_{4}^{\infty} e^{-4 x} d x\), which is less than \(0.0000001\). Equation Transcription: Text Transcription: integral _0 ^infinity e^-x^2 dx integral _0 ^4 e^-x^2 dx integral _4 ^infinity e^-x^2 dx n=8 integral _4 ^infinity e^-4x dx 0.0000001

The Laplace Transform If is continuous for , the Laplace transform of is the function defined by and the domain of is the set consisting of all numbers for which the integral converges. Find the Laplace transform of each of the following functions. (a) (b) (c)

The Laplace Transform If is continuous for , the Laplace transform of is the function defined by and the domain of is the set consisting of all numbers for which the integral converges. Show that if for , where and are constants, then the Laplace transform exists for .

The Laplace Transform If is continuous for , the Laplace transform of is the function defined by and the domain of is the set consisting of all numbers for which the integral converges. Suppose that and for , where is continuous. If the Laplace transform of is and the Laplace transform of is , show that

If is convergent and and are real numbers, show that

Show that \(\int_{0}^{\infty} x^{2} e^{-x^{2}} d x=\frac{1}{2} \int_{0}^{\infty} e^{-x^{2}} d x\). Equation Transcription: Text Transcription: integral _0 ^infinity x^2e^-x^2 dx=½ integral _0 ^infinity e^-x^2 dx

Show that \(\int_{0}^{\infty} e^{-x^{2}} \ d x=\int_{0}^{1} \sqrt{-\ln y} \ d y\) by interpreting the integrals as areas.

Find the value of the constant for which the integral \(\int_{0}^{\infty}\left(\frac{1}{\sqrt{x^{2}+4}}-\frac{c}{x+2}\right) d x\) converges. Evaluate the integral for this value of . Equation Transcription: Text Transcription: integral _0 ^infinity (1/square root x^2+4 - C/x+2) dx

Find the value of the constant for which the integral \(\int_{0}^{\infty}\left(\frac{x}{x^{2}+1}-\frac{C}{3 x+1}\right) d x\) converges. Evaluate the integral for this value of . Equation Transcription: Text Transcription: integral _0 ^infinity x^2e^-x^2 dx=½ integral _0 ^infinity e^-x^2 dx

Suppose is continuous on and . Is it possible that is convergent?

Show that if \(a>-1\) and \(b>a+1\), then the following integral is convergent. \(\int_{0}^{\infty} \frac{x^{a}}{1+x^{b}} \ d x\)