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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Proving $\int_{-\infty }^{+\infty }\frac{1}{(e^x-x)^2+\pi ^2}dx=\frac{1}{e^{-e^{-e^{-e..}}}+1}$

$$\int_{-\infty }^{+\infty }\frac{1}{(e^x-x)^2+\pi ^2}dx=\frac{1}{e^{-e^{-e^{-e..}}}+1}$$ I saw this integral in a mathematical book but I don't know how to find this result.
E.H.E
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Integral of $x(1-x^2)^n$

This should be "simple" according to the book but I can't seem to work it out. $n$ is an integer. $\int_0 ^1 x(1-x^2)^n dx$ I have tried binomial expansion and get stuck at $\sum_0^n {n\choose k}(-1)^k \frac{1}{2k + 2}$. I tried summing this by…
Mark
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Does the Riemann integral of the following function equal 0 ?

Does the Riemann integral of the following function equal $0$? $$ f(x) = \left\{ \begin{array}{ll} 1 , & \hbox{if } x=\frac{1}{n};\\ 0 , & \hbox{otherwsie.} \end{array} \right. $$ I need to prove it is integrable; now I can do that…
YNWA
  • 509
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Is the following function Riemann integrable?

Consider $f:\mathbb{R^2}\to\mathbb{R}$ defined by: $$f(x,y):=\arctan\frac{1}{x-y}\quad\forall x\neq y$$ $$f(x,x):=0$$ Is the function Riemann integrable in the square $[0,1]\times[0,1]$? I just have no clue how to attack this, surely basic,…
F.Webber
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How prove this integral $\int_{0}^{1}\frac{(x-1)dx}{(x+1)\ln{x}}=\ln{\frac{\pi}{2}}$

show that $$I=\int_{0}^{1}\dfrac{(x-1)dx}{(x+1)\ln{x}}=\ln{\left(\dfrac{\pi}{2}\right)}$$ I say this integral background,today I have use this wolf play some function,and Suddenly I found this interesting problem I try solve this by…
math110
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How do you integrate the function $\frac 1{x^2 - a^2}$, where a is a constant?

My problem with integrating by parts is that it always ends up being recursive, as I feel like I'm going in loops. Would appreciate it if someone can help me understand the process. $$\int \dfrac 1{x^2-a^2}dx$$ I know the answer is supposed to be…
user1746848
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Show that $\int_0^\infty t e^{-xt}\log(1+x^2) \, dx=(\pi-2\mathrm{Si}(t))\sin t-2\mathrm{Ci}(t)\cos t$, $\forall t>0$.

How to show that $$\int_0^\infty t e^{-xt}\log(1+x^2) \, dx=(\pi-2\mathrm{Si}(t))\sin t-2\mathrm{Ci}(t)\cos t$$ for all $t>0$?
Rotman
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evaluation of this logarithmic integrals

what is the value $$ \int_{a}^{\infty} \frac{\log^{n}(x)}{x^{2}}\mathrm{d}x $$ 'a' is a positive integer and so is 'n' my gues with a change of variable $ x=e^{t} $ is that this integral would be related to the incomplete gamma function $$ \Gamma…
Jose Garcia
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Find the equation of a curve where $\frac{dy}{dx}=2x+y$ at all points

Find the equation of the curve which passes through the origin and is such that $\frac{dy}{dx}=2x+y$ at all points $(x,y)$ on the curve, giving the equation in the form $y=f(x)$. I checked with WolframAlpha, and the first step listed was to…
hohner
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Stuck with an integration

How do you integrate $$ \int_0^1 y^{k_1} (1 - (1-y)^{\alpha})^{k_2}\,\mathrm{d}y $$ where $k_1$ and $k_2$ are non-negative integers and $\alpha>1$?
Nischal
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Evaluate double integral $ \int _0^{\sqrt{\pi}} \int_y^{\sqrt{\pi}} \sin (y^2 )\; dydx$

$$ \int _0^{\sqrt{\pi}} \int_y^{\sqrt{\pi}} \sin (y^2 )\; dydx$$ Even if I change the order of integration I don't see how to get rid of this $\sin (x^2)$ which doesn't have antiderivate. It is possible to evaluate it without using any aproximation…
user115442
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How to take this integral: $\int_0^\infty e^{-w^2\cos ^2 x-x} \, dx$?

How to evaluate $$\int_0^\infty e^{-w^2\cos ^2 x-x} \, dx$$ I have no clue, indefinite integral also would be a help.
Anixx
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Computation of double integral $(1-xy)^b$

I want to determine the values of $b>0$ such that $$\int_0^1\int_0^1(1-xy)^{-b}dydx$$ exists and is finite. I think that the integral is finite for $0
user191208
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Question about an integral.. Why is $a = $2?

I found the item in monbukagakusho 2013 math B exam. Consider the function $$F(x) = \int_a^x f(t)\ dt = x^3 - 2x^2 - x - a$$ with $a \ne 0$. Find $a$. I looked at the answer sheet and $a = 2$, but I ended up with weird equation (?) $0 =…
Blue Tomato
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How can we solve the integral $\int_\epsilon^1 e^{x^n\log x }\mathrm{d}x$?

Set an $\epsilon\in(0,1)$. How can we solve the integral $\int_\epsilon^1 e^{x^n\log x }{\rm d }x$ for all $n\in\mathbb{N}$? My attempt. Set $I_n=\int_\epsilon^1 e^{x^n\log x }{\rm d }x$. I tried a recursive procedure to express $I_{n}$ in terms of…
Elias Costa
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