All aspects of integration, including the definition of the integral and computing different types of integrals. For questions solely about the properties of integrals, don't use this tag alone! Use in conjunction with (indefinite-integral), (definite-integral), (improper-integrals) or another ...

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99 views

How to prove that $\int_{-1}^{1}\exp\left(\frac{1}{x^2-1}\right) \ dx=1$?

I have some trouble to prove that $$\int_{-1}^{1}\exp\left(\frac{1}{x^2-1}\right) \ dx=1\ ? $$
2
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3answers
153 views

Integration by parts.

How can I integrate $ \int_{3}^8 \ln \sqrt{x+1}\ dx$ by parts ? Is this step right ? $ \int_{3}^8 \frac{1}{2}\ln(x+1)\ dx $ = $ \frac{1}{2} \int_{3}^8\ln(x+1)\ dx$ $f^{'}(x) = 1 , f(x) = x , ...
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1answer
32 views

How to find $\int_{S^2}f \cdot n \ \text{d}S$ if $f(x,y,z):=(x^3,y^3,z^3)^T$

With $\mathbb{S}^2$ being the unit sphere, how to find $$\int\limits_{\mathbb{S}^2} \vec{f} \cdot \vec{n} \ \text{d}S$$ if $\vec{f}(x,y,z):=(x^3,y^3,z^3)^T$? Apparently, we need to use Gauss. ...
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4answers
129 views

Calculate the value of $\int_0^\frac{\pi}{6} \frac{\cos x \operatorname d\!x}{\sqrt{\frac{1}{4}-\sin^2x}}$

$$\int_0^\frac{\pi}{6} \frac{\cos x \operatorname d\!x}{\sqrt{\frac{1}{4}-\sin^2x}}$$ so $$\lim_{\epsilon->\frac{\pi}{6}} \int^{\epsilon} _{0} \frac{\cos x}{\sqrt{\frac{1}{4} - \sin^2x }} $$ ...
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1answer
113 views

Find the surface area obtained by rotating $y= 1+3 x^2$ from $x=0$ to $x = 2$ about the $y$-axis

Find the surface area obtained by rotating $y= 1+3 x^2$ from $x=0$ to $x = 2$ about the $y$-axis. Having trouble evaluating the integral: Solved for $x$: $x=0, y=1$ $x=2, y=13$ $$\int_1^{13} ...
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2answers
61 views

Partial fractions: why does $\int dt \implies t + C$

I am working on a partial fraction problem here, I understand everything in the problem except $t+C$, so I'd like to know where did the $t+C$ come from ? I want to solve this integral $$ \int ...
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0answers
10 views

Existence of invers of a function from covariance matrices space

Let $(\mathbb{R}^k,{\cal A})$ be a measurable space. Fix $c>0$ and for every $X\in \mathbb{R}^k$ define $X_c$ as $k-$dimensional vector such that the $i-th$ element of $X^{(c)}$ is $(-c)\vee ...
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1answer
61 views

Evaluating $\int\frac{3x+1}{2x^2-2x+3}dx$

Sorry I don't know how to use MathJaX but i've got a problem here that nobody seems to be able to explain to me. $$\int{3x + 1 \over 2x^{2} - 2x + 3}\,{\rm d}x.$$ It seems rather simple at first ...
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2answers
65 views

General solution to the integral of 4/x

A friend asked me what was the solution to the problem (which was on his test)$$\int\frac4xdx$$ I proceeded to tell him that you can take out the 4 in the numerator, and then just take the integral of ...
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1answer
34 views

Invariance of the Haar measure — upon inverses?

To simplify, let's assume $\mu$ is an invariant Haar measure on a commutative locally compact group $G$. Then, this means that $\mu$ is invariant under translation $\mu(U)=\mu(aU)$. However, I ...
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2answers
64 views

Constant area under $e^{-Ax^2}$ for different $A$

I am trying to find a solution to calculate relationship between an amplitude and boundaries of a Gaussian function so that an area under the curve is constant, let's say 2. I found a solution via ...
2
votes
3answers
204 views

A tricky Definite Integral

What is the value of $$\int_{\pi/4}^{3\pi/4}\frac{1}{1+\sin x}\operatorname{d}x\quad ?$$ The book from which I have seen this has treated it as a problem of indefinite integral and then directly put ...
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0answers
39 views

Question about finding volume using integration?

The question is: Find the volume of the solid whose base is a circle $x^2 + y^2 = 81$ and the cross sections perpendicular to the $x-axis$ are triangles whose height and base are equal. Now what the ...
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votes
3answers
179 views

solving $\int x^7\sqrt{3+2x^4}dx$

I'm trying to solve $\int x^7\sqrt{3+2x^4}dx$ All I have so far is Let $u$ = $3+2x^4$ $du$ = $8x^3$ $dx$ $\frac{du}{8x^3}$ = $dx$ Therefore, $\int x^7\sqrt{u}$ $\frac{du}{8x^3}$ ...
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1answer
29 views

Appreciate help with solving a probability density function for its constant term

I am using StackOverflow a lot for asking and answering programming related questions, and I hope it is appropriate if I'd ask my question below on here on this sister-site. If not, please let me know ...
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1answer
27 views

Multivariable Calculus Surface Integral Calculation

I have a surface bounded by $x^2+y^2=1$ and $x^2+y^2=9$ (cylinders) as well as the planes z=0 and z=3.The vector field is $(yx^3,xy^3,x)$. I know this involves the divergence theorem, where I would ...
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0answers
55 views

Real Analysis Integration Question

Still working through some things that I don't quite understand. I think this will make considerably more sense once I'm actually enrolled in the course this summer. For now, self study it is... ...
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2answers
34 views

Three Surface Integrals

Could someone assist with the following three surface integrals? Q1 The portion of the cone $z=\sqrt{x^2+y^2}$ that lies inside the cylinder $x^2+y^2 =2x$. Q2 The portion of the paraboloid ...
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0answers
38 views

Need help evaluating $\lim\limits_{n \to \infty} \frac{1}{n} \int_1^n \Vert\frac{n}{x}\Vert dx$

$$ \mbox{Evaluate}\quad \lim_{n \to \infty}{1 \over n}\int_{1}^{n}\left\Vert\,n \over x\,\right\Vert \,{\rm d}x $$ Where $\left\vert\left\vert\, x\,\right\vert\right\vert : \mathbb{R} \to \mathbb{R}$ ...
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1answer
50 views

Multivariable Calculus Green's Theorem Along Two Separate Curves

I have a curve which starts on $(-2,0)$ then goes to $(2,0)$ along the curve $y=4-x^2$ then back to the point $(-2,0)$ along the curve $y=x^2-4$. I have to compute the line integral $$\int -4x^2y \ ...
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1answer
134 views

evaluate $\int\frac{3x}{\sqrt{1-2x}}dx$

I'm trying to evaluate $\int\frac{3x}{\sqrt{1-2x}}dx$ This is what I got so far: Let $u$ = $1-2x$ $x$ = $\frac{u-1}{-2}$ $du$ = $-2$ $dx$ $\frac{-du}{2}$ = $dx$ Therefore, ...
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2answers
89 views

Integral of $x^2e^{-ax^2}$

Hey guys I need to find the following integral using integration by parts and not the gamma function. Also there is an a constant a in the exponential function. So it is actually $x^2e^{-ax^2}$. ...
3
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1answer
46 views

Question about finding the volume of a Sphere to a certain point

I've done a few things but I cant seem to figure out how to solve this. Any help please?
3
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5answers
211 views

How to evaluate this Trig integral?

I need to find the definite integral of $\int(1+x^2)^{-4}~dx$ from $0$ to $\infty$ . I rewrite this as $\dfrac{1}{(1+x^2)^4}$ . The $\dfrac{1}{1+x^2}$ part, from $0$ to $\infty$ , seems easy ...
2
votes
2answers
80 views

Prove that $ \int \limits_a^b f(x) dx$ = $ \int \limits_a^b f(a+b-x) dx$

Hi everyone I have been trying to prove that that $ \int \limits_a^b f(x) dx$ = $ \int \limits_a^b f(a+b-x) dx$ . Heres my attempt: LS: $ \int \limits_a^b f(x) dx$ = $ \int \limits f(b) - \int ...
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1answer
57 views

Area of a sphere bounded by a paraboloid

I need to find the area of the surface $x^2+y^2+z^2 = a^2$ for $y^2 \ge a(a+x)$. I know that $A = 4a \int_{-a}^0 dx \int_{\sqrt{a^2+ax}}^{\sqrt{a^2-x^2}} \frac{dy}{\sqrt{a^2-x^2-y^2}}$, but I have ...
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1answer
59 views

Proof of PDF Integrals

Hi guys my professor gave us some sample proofs to try at home and I was having trouble with 4 of them. I figured out how to do part (a) by using polar coordinates but cannot wrap my head around the ...
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2answers
112 views

Computing $\int_{0}^{\pi} {\cos(x)\sin(2x)}dx$

I'm trying to compute the following Integral $\int_{0}^{\pi} {\cos(x)\sin(2x)}dx$ This is what i've got so far: $\int_{0}^{\pi} {\cos(x)\sin(2x)}dx =\int_{0}^{\pi} {\cos(x)2\sin(x)\cos(x)}dx = ...
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1answer
57 views

Evaluating integral by parts.

Evaluate the following integral. $$ \int e^{2x}\sin{5x}\ dx $$ What I have tried : $ g(x) = \sin5x , f^{'}(x) = e^{2x} , f(x) = e^{2x} $ $$ \int e^{2x}\sin{5x}\ dx = e^{2x}\sin{5x} -\int e^{2x}\ ...
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votes
2answers
49 views

How to integrate absolute function

I have this absolute e-function, but I don't know how to calculate the integration $$ \int_{-2}^{2} e^{\frac{1}{2}j\omega |x|}dx $$ Any idea?
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1answer
48 views

Evaluating an integral by parts.

Evaluate the following integral. $$ \int x^2 e^x\ dx $$ What i have tried : $ f^{'}(x) = e^x , f(x) = e^x , g(x) = x^2$ $$ \int x^2 e^x\ dx = e^x\ x^2 - \int e^x\ 2x\ dx $$ $ f^{'}(x) = e^x , ...
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0answers
81 views

How to show a curve has a bertrand mate? (differential geometry)

Suppose $Y$ is a $C^2-$ arc-length parametrized curve on the unit sphere. For any nonzero constant $\lambda$ and $0 <\theta< \frac{\pi}{2}$,define: $α(t)= \lambda (∫Y(s)ds+ ...
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1answer
33 views

I have some approximate integral calculation. Is there a clean way to prove it?

Let: $P(R)=\int_R^{\infty}F(z)e^{-z}dz$ where $F(z)$ is the CDF of some discreate positive R.V. denote by $U$. Integrate by parts: $P(R)=(-F(z)e^{-z})_R^{\infty}+\int_R^{\infty}F'(z)e^{-z}dz$ The ...
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2answers
122 views

Inverse Functions and $u$-Substitution

Back in my undergrad days I wrote a false proof of the following. Problem. Prove that $\displaystyle\int_0^{2\pi}\frac{dx}{1+e^{\sin{x}}}=\pi$ Proof. Integrating by parts gives $$ ...
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2answers
62 views

Question about apply integrals in finding volume of a pyramid?

The answers entered already is what I got but either one or both are wrong. Can someone help me solve this problem?
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votes
3answers
148 views

Properties of integrable function

Given: $f$ is Riemann integrable on $[a,b]$ and $f(x)\geq 0$ for all $x$. Prove that if \begin{equation} \int_a^b f(x) dx=0 \end{equation} and $f$ is continuous, then $f(x)=0$ for all $x$. My idea: ...
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2answers
68 views

Solving integral without fundamental theorem of calculus

Solve $\int_0^1 3xdx$ without using the fundamental theorem of calculus. I know that, to solve an integral without the fundamental theorem of calculus, I can find the upper sum and the lower sum. I ...
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votes
1answer
75 views

Integral of $\frac{x}{x}, \frac{2}{x}, \frac{x}{2}$, and how they relate.

I'm studying for my diploma of higher studies (i.e. the diploma which gives me access to university) and I have a bit of trouble with building intuiton around integrals. Derivatives were relatively ...
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1answer
70 views

Tips for integrating on a symmetric domain?

One of the problems of my homework consists in integrating $\iint_D(x^2y^2+sin(xy)e^{{x^2}y^2})dA$ on the quadrilateral domain $D$ formed by (1,0), (0,2), (-1,0) and (0,-2). This domain is symmetric ...
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1answer
249 views

Find an expression for the area under the graph of f(x) as a limit?

f(X) = 2x/(x^2 +1), 1 <= x <= 3 Basically, I need to find an expression for the area under the graph within these intervals for the function as a limit. I understand the concept of the area ...
2
votes
4answers
399 views

Find integral when $dx$ is in the numerator

Can someone please walk me through the steps to find the following integral? I'm not sure what to do when $dx$ is at the top. $$ \int \frac{ x^{2}dx }{ (x^{3} + 5)^{2}} $$
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0answers
65 views

Integrating With Respect To $x$

Suppose I have the first derivative of the function $y$, $\displaystyle \frac{dy}{dx} = g(x)$. Furthermore, suppose I want to obtain the function $y$ by integrating with respect to $x$: ...
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3answers
167 views

Center of Mass via integration for ellipsoid

I need some help with the following calculation: I have to calculate the coordinates of the center of mass for the ellipsoid $$\left( \frac{x}{a} \right)^2 + \left( \frac{y}{b} \right)^2 + \left( ...
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1answer
53 views

How can I find this integral

How can I find this integral? please help $$I=\int_0^1\frac{e^{-\sqrt{x}}}{\sqrt{x}}\ dx.$$
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1answer
58 views

Riemann Integral - Partitioning - Nets

Everybody Hello, I'm confused about the following integral: Consider the following: Riemann Integral: $\int f dx:=\{\sum_{E\in\mathcal{E}} f(E)\lambda(E)\}_\mathcal{E}$ Domain: $I:=[0,1]$ Function: ...
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3answers
190 views

Let $f :[0,\pi] \rightarrow \mathbb R$ be a continuous function such that $f(0) = 0$ . Which of the following statements are true?

1.If $$\int_0^{\pi} f(t)\cos(nt)~dt=0$$ for all $n\in\{0\}\cup\mathbb N$ , then $f\equiv0$ . 2.If $$\int_0^{\pi} f(t)\sin(nt)~dt=0$$ for all $n\in\{0\}\cup\mathbb N$ , then $f\equiv0$ .
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vote
1answer
57 views

integral with gaussian function

I am trying to evaluate the following integral: $$ \int_0^\infty{z^{m-1}\over\left[1+\left(\eta z\right)^n\right]^p}e^{-(z-b)^2\over c}\,{\rm d}z, $$ where the integration is w.r.t. to $z$, and the ...
0
votes
0answers
37 views

Computing an (iterated) integral

I am having problems to find a closed form depending on $n$ of the following integral. $$ \int_{t_{0}\ <\ w_{1}\ <\ \cdots\ <\ w_{n}\ <\ t}\ {{\rm d}w_{1}\cdots {\rm d}w_{n} \over ...
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0answers
36 views

Don't understand an integration by parts result involving a step function on spacetime domain

I'm reading this work. Let $\Omega$ be a bounded (open) domain, and define $Q=(0,T)\times\Omega$. For every $t \in [0,T]$, let $\Omega_1(t), \Omega_2(t)$ be open subsets of $\Omega$, with $S(t)$ the ...
0
votes
1answer
10 views

Integral over all possible 2-dimensional lines (maybe variance related)

I'm working on some image software and was hoping for any feedback on an integral that came up: $$M = \int_{\mathbb{L} \in \Omega} \int_{(x,y) \in \mathbb{L}} F'_L(x,y)^2$$ If there is a better way ...