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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show that $\int_{0}^{\infty}e^{-x}\ln(x)dx=-\gamma=\Gamma'(1) $

show that $$\int_{0}^{\infty}e^{-x}\ln(x)dx= -\gamma=\Gamma'(1)$$ where $\gamma$ is Euler–Mascheroni constant I started with $$\Gamma(z)=\left[ze^{\gamma z}\prod_{k=1}^{\infty}\left(1+\frac{z}{k}\right)e^{-z/k}\right]^{-1}.$$ by take log then…
mnsh
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Calculate a certain integral

Good day! Is this integral tabular? I calculated it in MatLab and am now trying to write down an analytical expression. How can I get a result? \begin{align} I &= \int\limits_{x=-\infty}^{-1} \frac{\mu \, dx}{2 \cdot (1+\mu^2 \cdot ((x-m) \cdot…
Антон
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Triple integral in spherical coordinate, where am I wrong

$$ \iiint_{D} z\left(x^{2}+y^{2}+z^{2}\right) \mathrm{d} x \mathrm{~d} y \mathrm{~d} z $$ D is given by $x^{2}+y^{2}+z^{2}\leq 2z$ I try to use $ \left\{\begin{matrix} x=r\sin \phi \cos \theta \\ y=r\sin \phi \sin \theta \\…
liyushu
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How to do the integral $\int_0^\pi\sin^{n}\theta\cos(a \cos\theta)\mathrm d\theta$?

I stuck into trouble when trying to solve the integral $$ \int_0^\pi\sin^{n}\theta\cos(a \cos\theta)\mathrm d\theta $$ where n is a positive integer, and $a>0$ is real number. Using Mathematica I found that when $n$ is odd the integral is easy to do…
PyroTechnics
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Motion blur integral

I was going through a biophysics paper and derived the main pieces of interest but ran into this integral:$$R=\frac{1}{\Delta t}\int \limits _0^{\Delta t}dt \int \limits _0^{\Delta t}dt's(t)s(t')\min (t,t')-\int \limits _0^{\Delta…
ramiro
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Using complex number in integration

I know, $$I_1 = \int \dfrac{dx}{\sqrt{x^2 - 1}} = \ln|x + \sqrt{x^2-1}| + c$$ But if I factor out $i$ from the denominator, I get: $$I_2 = -i \int \dfrac{dx}{\sqrt{1 - x^2}} = -i \sin^{-1}x + ic$$ Are these 2 expressions equivalent?
Shub
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Evaluate $\int \dfrac{dx}{(1+x)\sqrt{1+2x-x^2}}$

$$\int \dfrac{dx}{(1+x)\sqrt{1+2x-x^2}}$$ I completed the square: $\int \dfrac{dx}{(1+x)\sqrt{2-(x-1)^2}}$ And then substituted $\sqrt 2\sin θ = x-1$ which gives $\int \dfrac{dθ}{\sqrt2 \sinθ + 2}$ But now I'm stuck. Can someone please help?
Shub
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Integrating $\int_{}^{} \frac{f'(r)}{f(r)}\frac{1}{1+ln(\frac{a}{f(r)})}dr$

I am trying to find a solution for this integral: $\int_{}^{} \frac{f'(r)}{f(r)}\frac{1}{1+ln(\frac{a}{f(r)})}dr$ My professor hinted that guessing the solution is a good way to go, but I am unable to make the correct guess.
RandomAsker889
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Showing $\int_{0}^{2022}x^{2}-\lfloor{x}\rfloor\lceil{x}\rceil dx = 674$

It is from the 2022 MIT Integration Bee Question 3 states as follows: $$\int_{0}^{2022}x^{2}-\lfloor{x}\rfloor\lceil{x}\rceil dx$$ I know that the answer is $674$, but I do not know the process and the steps to derive this solution. Can someone…
Kyeton Frawley
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Integrating $\int \frac{dx}{\sin^2 x \cos^2x-6\sin x\cos x}$

Our goal is to find $$\int \frac{dx}{\sin^2 x \cos^2x-6\sin x\cos x}$$ Here is my approach: We divide both numerator and denominator by $\cos^2x$ and the letting $\tan x=u$,our integrand becomes $-\int \frac{u^2+1}{u(6u^2-u+6)} du$. It is do-able…
madness
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How to perform the integral $\int_{-\infty}^\infty\frac{e^{-x^2}\sin(x) }{x }$?

Does anybody know how to perform the integral $$ \int_{-\infty}^\infty\frac{e^{-x^2}\sin(x) }{x } $$ Thanks.
user86406
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Calculating $\int \dfrac{\cot^3x}{\sqrt{1+\csc^4x}}\;dx$

Evaluate:$$\int \dfrac{\cot^3x}{\sqrt{1+\csc^4x}}\;dx$$
Y-dog
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How can the sign of this integral-based function at a particular point be determined?

The function $g(x)$ is defined thusly: $$g(x) = \int_1^x (t^2 - 3t + 2) e^{-t^2} dt$$ Is it the case that $g(3) > 0$? This question is part of a sample quiz provided by a high school student I'm tutoring. The theme of the quiz is the Fundamental…
Sean
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Is $\int x^{dx}-1$ well defined?

My question is precisely the question here. A quick summary of the algebraic process to "solve" the integral follows (taken from miniprime1's answer): $\int x^{dx}-1 = \int \frac{x^{dx}-1}{{dx}} {dx} = \int (\lim_{h \to 0}\frac{x^h-1}{h}) {dx} =…
user905694
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What am I doing wrong with this Euler substitution?

I want to evaluate the integral $\displaystyle\int\frac{1}{x\sqrt{x^2+2}}dx$ using the substitution $\sqrt{x^2+2}=-x+t$. So I get from the substitution $x=\frac{t^2-2}{2t}$, $dx=\frac{t^2+2}{2t^2}dt$ and $\sqrt{x^2+2}=\frac{t^2+2}{2t}$. Substituting…
Pen and Paper
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