Mathematics Stack Exchange
Mathematics Stack Exchange

Questions tagged [integration]

For questions about the properties of integrals. Use in conjunction with (indefinite-integral), (definite-integral), (improper-integrals) or another tag(s) that describe the type of integral being considered. This tag often goes along with the (calculus) tag.

Integration is a major part of .

There are two main kinds of integrals:

  • definite integrals (e.g. proper and improper integrals), which often have numerical values
  • indefinite integrals, which group families of functions with the same derivative.

Several techniques to solve integrals have been developed, including integration by parts, substitution, trigonometric substitution, and partial fractions.

Integration can be used to find the area under a graph and find the average of the function. Also, it can be used to compute the volume of certain solids and to find the displacement of a particle.

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need example a Riemann-integrable function is not continuous

every continuous function is Riemann integrable,continuity is certainly not necessary. I dont know anything about measure.
jujuju
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Transforming a multiple intergral

The following equality seems to hold: \begin{equation} \int_0^\infty\cdots\int_0^\infty F(\sum_{i=1}^kx_i)\text{ d}x_k\cdots\text{ d}x_1=\int_0^\infty \frac{y^{k-1}}{(k-1)!}F(y)\text{ d}y. \end{equation} Is there an easy way to see this?
MthQ
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How to calculate the following double integral

I would appreciate if you could help me to find the following integral, thank you. $$f(u)= \int_{-\infty }^{\infty} \int_{-\infty }^{\infty} \frac{e^{ -(x-a)^2/2b^2} }{{b\sqrt {2\pi}}} \frac{e^{ -(y-c)^2/2d^2} }{{d\sqrt {2\pi}}} \delta (xy-u) dx…
May
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How prove this $1+q\int_{0}^{1}x^{1-qx}dx=\sum_{k=0}^{\infty}\left(\frac{q}{k+1}\right)^k$

show that: $$1+q\int_{0}^{1}x^{1-qx}dx=\sum_{k=0}^{\infty}\left(\dfrac{q}{k+1}\right)^k\cdots\cdots(1)$$ I kown prove following this $$\int_{0}^{1}x^{-qx}dx=\sum_{n=0}^{\infty}\dfrac{q^n}{(n+1)^{n+1}}$$ note…
math110
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Center of mass of semi-sphere

I haven't calculated center of mass before and I'd like to know how I can do it in practise. I want to find the center of mass of a semi-sphere. Could you explain me, step by step, what I have to do? Many thanks
sunrise
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integrals inequalities

$$ \left( {\int\limits_0^1 {f^2(x)\ \text{d}x} }\right)^{\frac{1} {2}} \ \geqslant \quad \int\limits_0^1 {\left| {f(x)} \right|\ \text{d}x} $$ I can't prove it )=
Daniel
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An integral to solve

I have some problems trying to solve this integral: $$ \ \int^\ \frac{(x^2-1)\,dx}{(x^4+3x^2+1)\cdot\arctan\left(\frac{x^2+1}{x}\right)^{1001}}\,. $$ I can see $$ (x^2+1)^2+(x)^2=x^4+3x^2+1 $$ So i´m thinking in do some trigonometrical…
TBRB
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Finding a suitable substitution for integral

I'm struggling to find a suitable substitution for this integral: $$ \int{\frac{\sqrt{x^2+4}}{x}} $$ I've tried $u=x^2+4$, $u^2=x^2+4$ and some trigonemetric identities, but not much progress. Can anybody help me figure out how to get to the…
hohner
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Find $f(n,r)$ in $\int \frac{1}{\prod_{r=0}^{n} (x+r)}\, dx=\sum_{r=0}^n \frac{(-1)^r}{f(n,r)}p(x)+K$

Find $f(n,r)$ in the below equation;$$\int \frac{1}{\prod_{r=0}^{n} (x+r)}\, dx=\sum_{r=0}^n \frac{(-1)^r}{f(n,r)}p(x)+K$$ Here, K is the constant of Integration. What I have tried so far; $$L=\int \frac{1}{\prod_{r=0}^{n} (x+r)}\, dx=\int…
NadiKeUssPar
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integrating $\int \sqrt{2-2\cos(x)} \, dx$

So i am having some trouble getting the solution to the integral: $\int \sqrt{2-2\cos(x)} \, dx$ i made my first substitution $u = 2-2\cos(x)$ $u' = 2\sin(x) \, dx$ then... $\int \dfrac{1}{\sqrt{4-u}} \, du$ then the next sub of $s = 4-u$ $s' =…
user90950
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Can someone explain how to integrate it? $\int\frac{(x^2+1)dx}{x\sqrt{x^4+3x^3-2x^2-3x+1}},x>1$

Can someone explain how to integrate it? $$\int\frac{(x^2+1)dx}{x\sqrt{x^4+3x^3-2x^2-3x+1}},x>1$$ I know that I should use a substitution but I can't figure out. I should make some transformations under the root.
James
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Compute $\int_{\mathbb{R}^n}|x|e^{-|x|^2}dx$ in $\mathbb{R}^n$

How to integrate $$\int_{\mathbb{R}^n}|x|e^{-|x|^2}dx$$ in high dimension $\mathbb{R}^n$? In one dimension, by change of variables $s=x^2$, we have \begin{equation} \int_{-\infty}^\infty |x|e^{-x^2}dx = 2\int_0^\infty xe^{-x^2}dx = \int_0^\infty…
Wang
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Curious integral

I am interested in computing a rather nasty integral of the following form: $$ 4\int_0^x \mathrm{d}\theta…
unexpectedvalue
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Integrating $ (\cos ^3 x - \sin^3 x)^{-1}$

$$I=\int \frac{dx}{\cos^3 x-\sin ^3 x}=\int \frac {(\cos x-\sin x) dx}{(\cos x -\sin x)^2 (1+\sin x \cos x)}=\int \frac{2(\cos x-\sin x)dx}{(1-\sin 2x)(2+\sin 2x)}$$ $$\implies I=\frac{2}{3} \int (\cos x - \sin x) {\left(\frac{1}{1-\sin 2x}+…
Z Ahmed
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How do i find the volume of this paraboloid using triple interal

I need to find the volume of this paraboloid, $x^2+y^2=z,z=4y,$ using triple integrals, but have trouble determining the limits of integration. I know it's an inverted paraboloid with the plane $z=4y$ cutting at top.
Aman Mittal
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