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.

73636 questions
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How to integrate $\int_{-\infty} ^\infty \frac{\cos(xy)}{x^2+1}dx$

Is there a standard trick to compute this integral for $y\ge 0$? $\int_{-\infty} ^\infty \frac{\cos(xy)}{x^2+1}dx = \int_{-\infty}^{\infty}\frac{y \cos(x)}{x^2+y^2}$ Hopefully the same trick could be used to evaluate $\int_{-\infty} ^\infty…
Mark
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Integral $\int^{ \pi /2}_{0} \ln (\sin x)\ dx$

$$\int^{ \pi /2}_{0} \ln (\sin x)\ dx$$ The answer is $- \frac{\pi}{2} \ln 2$. I have changed it into $$\frac{1}{2} \int^{1}_{0} t d \ln t^{2}$$ But I didn't get the answer with it.
Zhen Zhang
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How to evaluate this integral with $\sin$ and $\cos$?

$$ \int \sin^6 x\cos^4x\ dx$$ I have absolutely no idea how to solve it. I tried to use formula of degree reduction but failed. Help me please. Thank you.
uley
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Is $\frac{1}{\sqrt{2\pi t}} \int_{\mathbb{R}} e^{-\frac{(x-iut)^2}{2t}} \, dx=1$?

$i$ is such that $i^2=-1$. I am not familiar with complex integral. Is $$ \frac{1}{\sqrt{2\pi t}} \int_{\mathbb{R}} e^{-\frac{(x-iut)^2}{2t}} \, dx=1 $$ as if computing the probability of a normal density function despite the mean is imaginary.…
Tim
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Different methods of evaluating $\int\sqrt{a^2-x^2}dx$:

Is there a simple and nice way to solve $\int\sqrt{a^2-x^2}dx$: PS:I am not looking for a substitution like $x=a\sin p$,
Tom Lynd
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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 $\left\vert\left\vert\, x\,\right\vert\right\vert : \mathbb{R} \to \mathbb{R}$ denotes the [distance to the] closest integer to…
MT_
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Improper integral

I'd like some help to evaluate this integral : $$I=\int^\infty_0 \frac{x-1}{\ln(x)}\,e^{-x} \,dx$$ I tried to use parameter then I've got an integral of gamma function which I don't know how to integrate it . Any help will be greatly appreciate .
Lina
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How find this $\int_{1}^{2}f^2(x^2)dx+5\int_{2}^{3}f(x^2)dx+7\int_{3}^{4}f(x)dx=\dfrac{1871}{30}$

Determine all function $f:R\to R$ for which $$\int_{1}^{2}(f(x^2))^2dx+5\int_{2}^{3}f(x^2)dx+7\int_{3}^{4}f(x)dx=\dfrac{1871}{30}$$ show that $$f(x)=x?$$ because we easy to find this follow…
user94270
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Why can't you integrate over a discontinuity?

I'm asked to investigate this integral, $$\int_0^3 \frac{1}{(1-x)^2}\,\mathrm{d}x$$ I get -3/2, but the integrand has a discontinuity at $x=1$, so I can't just integrate it. What is the answer? And why do things go wrong when integrating over a…
user85798
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Evaluate $\int\frac{\cot{x}}{1+\sin{x}+\cos{x}} \mathrm dx$

Find this integral: $$\int\dfrac{\cot{x}}{1+\sin{x}+\cos{x}}\mathrm dx$$ My try:…
user94270
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Evaluating $\int_{0}^{\pi} \frac{\cos(nx)}{(p+\cos(x))^2+q^2}\ \mathrm dx$

I have a formula in my research, but have no idea how to get the explicit formula. $$\int_{0}^{\pi} \frac{\cos(nx)}{(p+\cos(x))^2+q^2}\ \mathrm dx$$ where n is an integer.
sam
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How is this integral related to the Golden Ratio?

I ran across an interesting integral and I am wondering how in the world it could relate to the Golden Ratio, $\frac{1}{\phi}$. The problem says the solution must include the Golden Ratio,…
Cody
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Integrating $\int\tan\theta\sec^5\theta\ d\theta$

This problem is relatively straight forward, but for some reason, my answer is off by the power of 1. $$\int \tan \theta \sec^5\theta d\theta $$ The steps I take are Step 1. $$ u = \sec \theta $$ $$ du = \tan\theta $$ Step 2. $$ \int u^5 du…
ConfusingCalc
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Strangeness when using different u's in u substitution to solve an integral

I must be missing something here, but I feel like I'm getting two different solutions when I try to solve the following integral using different u values in the u substitution method: $$\int \frac{1}{300+2t} dt$$ When I pull 1/2 out of the integral…
Dan
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An exotic integral

Good evening, We were playing with a friend on Desmos, and we came to $x \longmapsto \arctan\left(\exp\left(-\displaystyle\frac{1}{\sqrt{1-x^2}}\right) \right)$. Here is the graph : And we have : $$\int_{-1}^1…
LexLarn
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