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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Evaluating $\int_{0}^{\infty}\frac{1}{x}\big (\frac{\sinh ax}{\sinh x}-ae^{-2x}\big )dx$

Some time ago, stumbled out of an integral: $$\int_{0}^{\infty}\frac{1}x{}\left (\frac{\sinh ax}{\sinh x}-ae^{-2x}\right )dx=\ln\frac{\pi\cos\frac{a\pi}{2}}{\Gamma^2(\frac{a+1}{2})};\left | a \right |<1$$ I have no idea where to start?
Martin Gales
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How to evaluate the integral $\int_{0}^{\infty}\frac{\sin{(ax)}}{x^b} dx$

Evaluate the integral $$\int_0^\infty \dfrac{\sin(ax)}{x^b} \, dx,\quad a\in \mathbb{R},\quad 0
math110
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How to calculate this Integral which function is defined by parts

Let $k:[0,1]\times [0,1] \to \mathbb{R}$ defined as: $k(x,y) = \left\{ \begin{array}{ll} y(1-x) & \mbox{if } 0\leq y\leq x \le 1,\\ x(1-y) & \mbox{if } 0\leq x\leq y \le 1. \end{array} \right.$ If we denote $u_n(x):=\sin(n\pi x), \forall…
Blacks
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Sources for reading about integration techniques

I wanted to know about techniques like Contour Integration, Leibniz Rule and so on. Complex analysis textbooks, like other analysis textbooks, only discuss generalisations and simple integration using learned material. I came across the Leibniz rule…
Shimura Variety
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What is the integral of $\int e^x\,\sin x\,\,dx$?

I'm trying to solve the integral of $\left(\int e^x\,\sin x\,\,dx\right)$ (My solution): $\int e^x\sin\left(x\right)\,\,dx=$ $\int \sin\left(x\right) \,e^x\,\,dx=$ $\left(\sin(x)\,\int e^x\right)-\left(\int\sin^{'}(x)\,\left(\int…
Mohammad Fakhrey
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Integral of $L^2(\mathbb R^+)$ function is $o(\sqrt x)$

I stumbled upon an exercise which goes as follows : Let $f \in L^2(\mathbb R^+)$, show that $\int_0^x f(t)\text{d}t = o\left(\sqrt x\right)$. Cauchy-Schwarz inequality gives the fact that $\int_0^x f = O\left(\sqrt x\right)$ For decreasing…
B. Pillet
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Integration by parts involving the delta function

Often in physics we integrate by parts $$\int_{x_0}^{x_1} f(x) \frac{d}{dx}( \delta(x-y))dx$$ by: $$=[f(x) \delta(x-y)]_{x_0}^{x_1} - \int_{x_0}^{x_1} \delta(x-y) \frac{df}{dx} dx$$. I have a really simple question, how can we assume that $[f(x)…
Alex Gower
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Solution integral $\;\displaystyle \iint \sqrt{\cos^2(x \pi)+\sin^2(y \pi)} \ dx\,dy$

Working on a hobby project: "Circle from (2D) random walk" [SE] and came across this integral: $$\bar{R}=\int_0^1 \int_0^1 \sqrt{\cos^2(X \pi)+\sin^2(Y \pi)} \ dX\,dY$$ My intention is to have the mean vector length of every vector (starting in…
Vincent Preemen
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Integral of $\int 1/x dx$

Is this a sufficient proof for this integral?: $$\int \frac{dx}{x} = \ln |x| + \mathcal{C}$$ Let $$x = e^{u} : $$ $$\int \frac{dx}{x} = \int du = u + \mathcal{C} = \ln |x| + \mathcal{C}$$ I'm not sure :S I don't know if my logic's a bit wishy washy/…
user78416
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Integration exercise: $ \int \frac{e^{5x}}{ (e^{2x} - e^x - 20) }dx$

I have trouble integrating: $$ \int \frac{e^{5x}}{e^{2x} - e^x - 20} dx$$ With $t=e^x$, I've rewritten it as: $$\int \frac{t^5}{t^2 - t - 20} \frac{1}{t} dt$$ Then I tried integration by parts, but I am not any closer to the solution.
klaufir
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Find the limit using Riemann sum

Find the limit: $ \lim\limits_{n\to\infty}\sqrt[n]{\left(\dfrac{1 + n}{n^2} \right)\left(\dfrac{4 + 2n}{n^2} \right)...\left(\dfrac{n^2 + n^2}{n^2} \right)} $ I tried simplifying this limit and the one I get to…
A. Mason
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Stokes' Theorem and Measure Zero Sets

This is probably a very naive question but I am trying to connect two pieces of information in my head regarding integration of differential forms and integration with respect to a measure. The first piece is that Stokes' theorem implies the…
firemind
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Integral of $\int^1_0 \frac{dx}{1+e^{2x}}$

I am trying to solve this integral and I need your suggestions. I think about taking $1+e^{2x}$ and setting it as $t$, but I don't know how to continue now. $$\int^1_0 \frac{dx}{1+e^{2x}}$$ Thanks!
Ofir Attia
  • 3,136
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Evaluating $\int \cos^4(x)\operatorname d\!x$

I want to Evaluate integral :$$\int \cos^4(x)\operatorname d\!x$$ And think about doing the following thing: $$ \int \left(1-\sin^2(x)\right)^2\operatorname d\!x \to \int \left(1-2\sin^2(x)+\sin^4(x)\right)\operatorname d\!x $$ but I think I just…
Ofir Attia
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How to use Stokes theorem?

Use Stokes theorem to evaluate $\int_{C} [ydx+y^{2}dy+(x+2z)dz]$ where $C$ is the curve of intersection of the sphere $x^{2}+y^{2}+z^{2}=a^{2}$ and the plane $y+z=a$ oriented counterclockwise as viewed from above. I can't seem to understand the…
user70337
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