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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Find the smallest value of the function $F:\alpha\in\mathbb R\rightarrow \int_0^2 f(x)f(a+x)dx$

Let $f -$ fixed continuous on the whole real axis function which is periodic with period $T = 2$, and it is known that the function $f$ decreases monotonically on the segment $[0, 1]$ increases monotonically on the segment $[1, 2]$ and for all $ x…
Roman83
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If $f(x) = ax^2+bx+c$ and $f(0) = 0$ and $f(2) = 2\;,$ Then minimum value of $\int_{0}^{2}|f'(x)|dx$

Consider the polynomial $f(x) = ax^2+bx+c$ and $f(0) = 0$ and $f(2) = 2\;,$ Then Minimum value of $\displaystyle \int_{0}^{2}|f'(x)|dx$ $\bf{My\; Try::}$ From $f(0) =0\;,$ We get $c=0$ and $f(2) = 2\;,$ We get $4a+2b=2\Rightarrow 2a+b=1$ So we get…
juantheron
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Evaluation of Definite Integral

Evaluation of $\displaystyle \int_{0}^{\frac{\pi}{2}}\frac{\cos x\sin 2x \sin 3x}{x}dx$ $\bf{My\;Try::}$ Let $\displaystyle \int_{0}^{\frac{\pi}{2}}\frac{\cos x\sin 2x \sin 3x}{x}dx = \frac{1}{2}\int_{0}^{\frac{\pi}{2}}\frac{2\sin 3x\cos x\sin…
juantheron
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Ramanujan-type identity $\sum_{n=1}^{\infty}\frac{n^3}{e^{2^{-k}n\pi}-1}=\sum_{n=0}^{k}16^{n-1}$

On my previous page on Ramanujan's-type identity Jack D'Aurizio and Paramanand Singh independently offered their own method of proving that beautiful identity. I am greatly appreciated for their efforts in proving the identity. Here we offered two…
user335850
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Evaluation of $\int_{0}^{1}\frac{1}{\sqrt{1+x}+\sqrt{1-x}+2}dx$

Evaluation of $$\int_{0}^{1}\frac{1}{\sqrt{1+x}+\sqrt{1-x}+2}dx$$ $\bf{My\; Try::}$ Let $$I = \int_{0}^{1}\frac{1}{\sqrt{1+x}+\sqrt{1-x}+2}dx$$ Put $x=\cos 2 \theta\;,$ Then $dx = -2\sin 2 \theta d\theta$ and Changing Limit, We get $$I =…
juantheron
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Evaluation of $\int_{0}^{1}\frac{x^4(1-x^2)^5}{(1+x^2)^{10}}dx$

Evaluation of $\displaystyle \int_{0}^{1}\frac{x^4(1-x^2)^5}{(1+x^2)^{10}}dx$ $\bf{My\; Try::}$ Let $$I = \int_{0}^{1}\frac{x^4(1-x^2)^5}{(1+x^2)^{10}}dx$$ Now Put $x=\tan \theta\;,$ Then $dx = \sec^2 \theta d\theta$ and changing limits, We…
juantheron
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Relation between $I_{k}$ and $I_{k+2}$

If $\displaystyle I_{k}=\int_{0}^{\frac{\pi}{2}}x(\sin x+\cos x)^ndx\;,n\in \mathbb{N}$ Then Relation between $I_{k}$ and $I_{k+2}$ $\bf{My\; Try::}$ Given $\displaystyle I_{k}=\int_{0}^{\frac{\pi}{2}}x(\sin x+\cos x)^ndx\;,$ Then $\displaystyle…
juantheron
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How to get to the closed form of $\int_{-\infty}^{\infty} \frac{x^2e^x}{(e^x+1)^2}$

I came across this integral when helping some friends with a statistical mechanics assignment, Mathematica reports it as $\frac{\pi^2}{3}$. So far I have noticed that the integrand is an even function so the integral is equivalent to,…
mike van der naald
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Dimensional analysis of $\int x \cos x dx$

I'm confused about using dimensional analysis, as in Street-Fighting Mathematics, of integrals like $$ \int \! x \cos x \, \textrm{d}x = \textrm{something} $$ I start by expressing the right side as a function of one or more variables. It seems…
Chewers Jingoist
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Evaluation of integral of product of three functions

How can I evaluate $\int_0^\infty x^2e^{-kx}\cos (kx)dx$ where $k>0$ ? I tried Laplace transforms and integration by parts. Those methods did not work. Help solicited.
Purushothaman
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How to integrate $\int \frac{ev+f}{av^2 + bv +c} dv$?

I want to integrate \begin{align} \int \frac{ev+f}{av^2 + bv +c} dv \end{align} can you give me some hints or detail procedure for this integral? From mathematca, i have \begin{align} \frac{-\frac{2 (b e-2 a f) \text{ArcTan}\left[\frac{b+2 a…
phy_math
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Does $\frac{1} { ax+b}$ apply when $b$ equals $0$

This is a dumb question but I'd still like to confirm it here, I just haven't found any information about this in internet. According to any table of integrals $\int \frac{1}{ax+b} \, dx = \frac{1}{a} \ln(ax+b)$ . This doesn't work if if $b=0$,…
kitchenisrip
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Riemann-Stieltjes integral and applications(Integration by parts ?)

Let $f,\phi:[a,b] \rightarrow \mathbb{R}$ be a continuous map, and the function of bounded variation respectively. And, $g$ is a continuous map on $[a,b]$. Then, following results hold. (i) A map $ \psi:[a,b] \rightarrow \mathbb{R}$ s.t.…
Chris kim
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Integral of $e^{(x-x^3)/3n}$ from $0$ to $\infty$

How can you compute the following integral assuming $n>0$? $$\int_{x=0}^{\infty}e^{\frac{x -x^3}{3n}}dx $$ Mathematica etc. fail to produce anything useful. EDIT: I would be happy with an asymptotic result in $n$ if it is too hard to compute…
user35671
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Antiderivative problem

What is the antiderivative of $(2x+7)^{1/2}$? My understanding is that it would be $\frac 23 \times(2x+7)^{3/2}$ but according to the source I am working from the answer is $\frac 13 \times (2x+7)^{3/2}$.
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