Concerns all aspects of integration, including the integral definition and computational methods. For questions solely about the properties of integrals, use in conjunction with (indefinite-integral), (definite-integral), (improper-integrals) or another tag(s) that typically describe(s) the types of ...

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2answers
75 views

How do I evaluate this integral $I = \int_{0}^{2 \pi} \ln (\sin x +\sqrt{1+\sin^2 x}) dx$?

I used some variables change to evaluate this integral but i'm not succeed may I have some wrong step as trigono-transformation.Then Is there some one who can show me how do evaluate this : $$I = ...
0
votes
2answers
58 views

Find the area of the entire region that lies between $r=1+\sin\theta; r=1+\cos\theta$

I have to find the area of the region that lies between the curves $r=1+\sin\theta; r=1+\cos\theta$ . The answer the book gave was $\frac {3\pi}{2}-2\sqrt{2}$ . I tried generating the curve for ...
0
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1answer
20 views

Derivative of an integral over a varying domain

Consider the function $$H(\alpha) = \int_{\Omega(\alpha)} h(\alpha,x) dx,$$ where $\alpha\in\mathbb{R}$ and $\Omega(\alpha)\subset\mathbb{R}^2$ is a domain that varies continuously with $\alpha$. Is ...
1
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1answer
25 views

Error bound of midpoint rules with unbounded second derivative

It is well known that error bound of midpoint rule for function $f[a,b]$ is given by $$ E\leq K\frac{(b-a)^3}{24 n^2} $$ where $|f(x)''\leq K|$ and $n$ is the number of time steps. if second ...
9
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3answers
99 views

What's happening at $a=-1$ in $\int x^a dx$? [duplicate]

If we take the right limit $$\lim_{a\to-1}\int x^a dx=\lim_{a\to-1}\frac{x^{a+1}}{a+1}=+\infty$$ but on the other hand $$\int\lim_{a\to-1} x^a dx=\ln x$$ I'm aware you can't just commute the ...
4
votes
1answer
68 views

Area under tangent to a curve.

The tangent to the graph of the function $y=f(x)$ at the point with abscissa $x=a$ forms with the line $x$-axis an angle $\frac{\pi}{6}$ and at the point with abscissa $x=b$ an angle of ...
1
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2answers
68 views

Integral of $x^2\sqrt{5+x}\ dx$

I have the following integral to solve, with my working out below. This is a bit more complicated than I am used to, so I'm hoping for some feedback as I'm not sure if my process & solution are ...
2
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3answers
59 views

derivative integral $\int_0^{x^2} \sin(t^2)dt$

I want to know how I derivative this integral: $$\int_0^{x^2} \sin(t^2)dt$$ what are the steps to derivative it?
4
votes
1answer
69 views

Find all functions such that $\int f(x)g(x) dx =\left(\int f(x) dx\right)\left(\int g(x) dx\right)$

Is it possible to find all functions such that $$\int f(x)g(x) dx =\left(\int f(x) dx\right)\left(\int g(x) dx\right)$$? My teacher asked us to give examples to prove that this is not true but I was ...
0
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0answers
36 views

Show the integration with a complex variable

I want to show that there exists inverse Laplace transform, $f(t)$ of the function $F(\lambda)$. In other word, given $F(\lambda)$, existence of function $f(t)$ such that $$ ...
2
votes
2answers
62 views

Finding $\int_{0}^{\frac{\pi}{2}}\frac{1}{\cos (x-\frac{\pi}{3}).\cos (x-\frac{\pi}{6})}\mathrm{d}x$

How can I find $$\int_{0}^{\frac{\pi}{2}}\frac{1}{\cos (x-\frac{\pi}{3}).\cos (x-\frac{\pi}{6})}\mathrm{d}x$$ ? I suspect this has something simple to do with the basic definite integral properties; ...
0
votes
3answers
59 views

Use the comparison test to find whether $\int_0^\infty 1/(x^2+1)^2\,dx$ converges or not

I was thinking what function I should compare it to. If I say whether a function is smaller or bigger than this one, then I must prove that. I was thinking of (x+1)^2 but I realized that this ...
2
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1answer
59 views

Find the area bounded between $f(x)=\frac{\arctan(x)}{x^2}$ and $g(x)=\frac{\arctan(x)}{x^2+1}$

Find the area bounded between $$f(x)=\frac{\arctan(x)}{x^2} \quad\text{and}\quad g(x)=\frac{\arctan(x)}{x^2+1}.$$ The title says the question. The limits are from 1 to infinity. I know that I ...
1
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1answer
48 views

Find the SA of a torus

I have been trying to find the surface area of the torus generated by the rotation of $(x-R)^2 + y^2 = r^2$ about the y axis. I tried to use the equation: $$\int_a^b2\pi y\sqrt{1+\left(\frac ...
14
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1answer
210 views

Evaluating $\int{ \frac{x^n}{1 + x + \frac{x^2}{2} + \cdots + \frac{x^n}{n!}}}dx$ using Pascal inversion [duplicate]

(Note: I apreciate very much who marked this as a duplicate but I would like an answer for why my proof is wrong) This is my solution, I have no clue why it failed. Let's start: define $$I_n(m) = ...
1
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2answers
78 views

Finding $\int_{0}^{e^2}(\frac{1}{\log{x}}-(\frac{1}{\log x})^2).\mathrm{d}x$

Finding $$\int_{0}^{e^2}(\frac{1}{\log{x}}-(\frac{1}{\log x})^2).\mathrm{d}x$$ I came upon this problem online, and the answer is given to be $(\frac{e^2}{2}) - e$. However, Wolfram Alpha states that ...
3
votes
1answer
42 views

Probability of True Positive of a random variable defined by an integral expression

$\newcommand{\Prob}{\operatorname{Prob}}$Let's assume that we have a random variable with the following pdf: \begin{equation} f_T(x) = \int_0^\infty f_T(x,g) \cdot f_{g}(g) \, dg = \int_0^\infty ...
2
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3answers
68 views

$I_{2n}=\dfrac{1\times 3\times \ldots \times (2n-1)}{2\times 4\times \ldots\times 2n}\times\dfrac{\pi}{2}\quad \forall n\geq 1$

let $$I_n=\int_0^{\frac{\pi}{2}}\cos^n(t) \, dt$$ show that $$I_{2n}=\frac{1\times 3\times \ldots \times (2n-1)}{2\times 4\times \ldots\times 2n}\times\dfrac{\pi}{2}\quad \forall n\geq ...
0
votes
0answers
25 views

Volume bounded by $y = \sqrt{25-x^2}, y=0, x=2, x=4$ about the $x$-axis [closed]

I have come up with this integral $$2\pi \int_3^{\sqrt{21}} y\left(4-\sqrt{25-y^2} \right)\, \mathrm{d}y + 2\pi \int_0^3 2y\, \mathrm{d}y$$ ...
2
votes
2answers
51 views

derivative of a function including a vector

given a column vector including function of a parameters $x=\bigg(f(\beta_1),\ldots,f(\beta_m)\bigg)^T$ where $T$ denotes transpose of the vector. Can somebody tells me what is the derivative with ...
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votes
0answers
48 views

Find a triple integral using spherical coordinates.

Solve the following integral: $$\iiint_{V} \mathrm{d}x\: \mathrm{d}y \:\mathrm{d}z$$ Where $V$ is part of the sphere $x^{2}+y^{2}+z^{2}=5$ which is above the $xy$ plane and inside the $x^2+y^2=1$ ...
1
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1answer
17 views

Evaluating the volume of a torus formed by rotating a region about a horizontal axis using shells.

Using the method of cylindrical shells, find the volume of the shape created by revolving the region $x^2+(y-5)^2=4$ about $y=-1$. A cylindrical shell is given by: $2\pi v f(v) \ dv$ I solve ...
12
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1answer
206 views

Calculating 2 integrals in polylogarithmic functions

Are we aware of any nice way of calculating these $2$ integrals? $$i) \space \int_0^1 \frac{\text{Li}_2\left(x-x^2\right)}{x^2-x+1} \, dx$$ $$ii)\space \int_0^1 ...
-2
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0answers
49 views

Is there a way to solve this integral? $\int_0^4 \frac {120x^ky^{4-k}}{\Gamma(5-k)\Gamma(1+k)}dk$ [closed]

Is there a way to evaluate this integral? $$\int_0^4 \frac {120x^ky^{4-k}}{\Gamma(5-k)\Gamma(1+k)}dk$$
2
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2answers
27 views

$F=w(x)=\frac{k}{x^2}$ How much work required to lift a satellite an “infinite distance” into outer space?

The satellite is 6000 lbs at earth's surface, a distance $R$ from the earth's centre (so the answer will be in terms of $R$). I know that it's supposed to be an improper integral going from 0 (?) to ...
0
votes
0answers
13 views

Proof of sets to integrate over in Green's, Stokes and Divergence Theorems

As the title says I've some doubts about these theorems, because my definitions of them are so abstract, I'm struggling to apply them. For Green's theorem, in order to be able to apply it, the region ...
0
votes
1answer
15 views

Evaluate the volume of a solid of revolution using shells.

A cylindrical shell $S$ formed by some revolution about the $y$-axis is given by the equation: $S=2\pi x f(x)dx$, where the circumference $C$ of the shell is $C=2\pi x$, the height of the shell ($H$) ...
1
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3answers
69 views

Volume of the region outside of a cylinder and inside a sphere

The cylinder is $x^2 +y^2 = 1$ and the sphere is $x^2 + y^2 + z^2 = 4$. I have to find the volume of the region outside the cylinder and inside the sphere. The triple spherical integral for this ...
1
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1answer
49 views

Integration Of exponential Function

I have tried almost everything, but can't solve this integral. $$\int e^{-1/x^2} \, dx $$
0
votes
1answer
57 views

About odd functions and improper integrals e.g. $\int^{\infty}_{-\infty}\sin x \; dx$

Does $\displaystyle \int^{\infty}_{-\infty}\sin x \; dx$ converge? Since $\sin x$ is an odd function, and we know that in definite integrals $\displaystyle \int^{a}_{-a}\sin x \; dx=0$ then does ...
3
votes
2answers
160 views

Evaluate an Integral

Evaluate: $$\int_{0}^{\infty}\dfrac{\sin^3(x-\frac{1}{x} )^5}{x^3} dx$$ I've been stumped by this Integral and cannot think of how to evaluate it. I substituted $\dfrac{1}{x^2}=t \Rightarrow ...
0
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5answers
82 views

Hint to finding $\int_{0}^{100} (x-\lfloor x\rfloor).\mathrm{d}x$

Wolfram Alpha gives it to be 50, and I seem to find no way to the solution. $$\int_{0}^{100} (x-\lfloor x\rfloor).\mathrm{d}x$$
1
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3answers
149 views

Find $\int_a^b \sin |x| \, \mathrm{d}x $

How to find the integral $$\int_a^b \sin |x| \, \mathrm{d}x \,?$$ I'm able to obtain definite integral of form $ \int_a^b \lvert\sin x \rvert \, \mathrm{d}x$ but not when the modulus operator is ...
1
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1answer
32 views

Help in understanding a limit involving an integral.

$$ I_n = \int^1_0\frac{x^n}{ax+b}dx $$ Where: $n \in N$ ; $a,b \in (0,\infty)$" Find $\Xi$, where: $$\Xi=\lim_{n \to \infty}nI_n$$
1
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1answer
63 views

Evaluate $\int_{0}^{\frac{\pi}{4}}\frac{\sec^2 \theta }{(1-\tan \theta )}\ d \theta$

Evaluate $$\int_{0}^{\frac{\pi}{4}}\frac{\sec^2 \theta }{(1-\tan \theta )}\ d \theta$$ Here's my attempt: $$u=1-tan \theta \implies -du=\sec^2 \theta d \theta$$ Substituting back in, I get ...
0
votes
1answer
28 views

Question about negative value using the ratio convergence test for integrals

Find for what $p$, $\displaystyle \int ^{\infty}_0 x^p \arctan x dx$ converges. By parts, it's equal to: $\displaystyle \lim_{b\to \infty}\frac 1 {p+1}x^{p+1}\arctan x |^b_0- \int ^b _0\frac ...
0
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1answer
30 views

Volume bounded by the regions $y= \frac{1}{x}, x=1, x=2, y= 0$ about $x= 3$ using the shell method

Find volume by these bounded regions $y=\dfrac{1}{x}, x=1, x=2, y= 0$ about $x= 3$ (shell method) Not sure what is wrong with my integral here. $$2\pi \int_1^2 \frac{(3-x}{x} \, \mathrm{d}x+ ...
0
votes
2answers
74 views

Integration problem: $\int \ln\left(\sin(\sqrt{x})+\cos(\sqrt{x})\right)dx $

I need help in solving the following problem: $$\int \ln\left(\sin(\sqrt{x})+\cos(\sqrt{x})\right)dx $$ I really don't know how to start solving this problem; any tips or solutions will be greatly ...
3
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1answer
59 views

Reference Book for Calculus

I've made this post as detailed as I can so as to give you a fair idea of my level. Calculus (particularly Integration) is my passion and frankly I spend all my free time learning as much of it as I ...
1
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2answers
88 views

Evaluate the indefinite integral $\int \frac{\cos \theta}{ \sqrt{2 - 9 \sin^2 \!\theta}} \mathrm{d}\theta$

I want to evaluate $$\int \dfrac{\cos \theta \, \mathrm{d}\theta}{ \sqrt{2 - 9 \sin^2 \theta}}$$ but I can't seem to get the answer, my working is as below:
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1answer
55 views

integration by part and a limit- Evans PDE Chapt2 problem 13

1) I am having a hard time in seeing how the integration by part done in this problem (page 11) enter link description here Could anyone help explaining? I cannot see how he got 3 terms instead of ...
0
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1answer
19 views

substitution rule and differentials

We use $\frac{dy}{dx}$ to represent derivative. In solving integrals we use substitution rule sometimes. That is $\frac{dy}{dx}$ $dx$ = $dy$ . But $\frac{dy}{dx}$ is not a fraction says my book. But ...
1
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2answers
135 views

find the limit for Integral

I tried to find the limit of this function, without any success, $$\lim _{x\to \infty }\frac{\displaystyle \left(\int _{10}^x\frac{e^{\sqrt{t}}}{t}dt\right)^2}{e^x}$$ I tried to solve it by ...
0
votes
2answers
47 views

Integrating inverse trigonometric functions

I want to find the integral of $$\frac {\sin^{-1}(\ln x)}{x}$$ I know the best way to find th integration of trigonometric shirt substitutions is to substitute to eliminate the inverse trigonometric ...
0
votes
1answer
83 views

Integration of $\frac{1}{\sqrt{x^8+1}} $ [closed]

How can we integrate $\frac{1}{\sqrt{x^8+1}}$ thanks for help. I couldn't find any way. I tried to factor, substituted with $\tan(a)$ but it's not working :'(
1
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1answer
19 views

Series Test for Integrability via the Distribition Function

I imagine that the following question has a well known (and perhaps, easily obtainable) answer, but I can't find it by myself nor along the references that I have in mind so far. So, if $f$ is a ...
4
votes
2answers
49 views

Complex integral with exponential and tangent

Suppose that $k \in \mathbb{R}.$ Evaluate as a function of $k$ the integral $$I(k) : = \int_{-\pi/2}^{\pi/2} e^{i \ k \ \mathrm{tan}(\phi)} d \phi.$$ Any suggestions on how to approach this problem? ...
0
votes
2answers
52 views

Integral with only a list of values

I am supposed to perform an integral of function $y(x)$ from say $x_1$ to $x_2$. Now the issue is I don't have an actual function $y(x)$, but I do have a list of values for $y$ and $x$. In what way ...
3
votes
0answers
71 views

When can we use Differentiation under the Integral sign?

Let me elaborate, 'Feynman's' trick ranks up in the top ten on most people's list, right behind contour integration, for best ways to evaluate definite integrals. However, unlike contour integration ...
-1
votes
2answers
38 views

Find the volume formed by rotating the region bounded by $y = e^{-x} \sin x$, $x\ge 0$ about $y =0$.

Find the volume formed by rotating the region bounded by $y = e^{-x} \sin x$, $x\ge 0$ about $y =0$. I tried to graph this using Wolfram Alpha, but it didn't help. I don't know how to start or ...