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 evaluate $\int_0^{\pi/6} \frac{x}{\sin x}$?

A well-know representation of the Catalan constant is given by $$C=\frac12\int_0^{\pi/2}\frac{x}{\sin x}\,dx.$$ So, I asked myself if there are other values of $x$ that make the integral $$f(x)=\int_0^x \frac{t}{\sin t}\,dt$$ take well-known closed…
Omran Kouba
  • 28,772
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Integrating $\int \frac{1}{\sqrt[4]{1+x^4}} dx $

How to integrate $$\int \frac{1}{\sqrt[4]{1+x^4}} dx $$
user91495
  • 51
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Finding the value of given integral.

It was asked to find the correct option(s) for the given integral: $$I_n = \displaystyle\int_{\frac{n}{2}}^{\frac{n+1}{2}}\dfrac{\sin{(\pi(\sin^2{\pi x}}))}{(\sqrt2)^x} \,…
Alan
  • 409
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Hard triple Integral $\int_{0}^{1}\int_{0}^{1}\int_{0}^{1}\frac{1}{2-zx^{2}-zy^{2}}dxdydz=\ln(2^{G})$

How do prove this triple integral? $$\int_{0}^{1}\int_{0}^{1}\int_{0}^{1}\frac{1}{2-zx^{2}-zy^{2}}dxdydz=\ln(2^{G})$$ where G is Catalan's constant. As my try I only reach to this hard single integral: $$\int…
asmatqatea
  • 142
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Anyone know if this integral has an analytic solution?

I've come across the following integral: $$\int_{-\pi}^{\pi}\left[\frac{1}{A-R \cos(2\theta-\phi)}\right]^{\frac{N-1}{2}}d\theta$$ I know how to approximate this integral using the Laplace method, just wondering if: a) Does this integral have an…
probabilityislogic
  • 1,055
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Calculate the integral of $\frac{1}{x^2 +x + \sqrt x}$

How to correctly calculate the integral: $$\int_0^\infty \frac{1}{x^2 +x + \sqrt x}dx$$ Edit: I tried to figure out if the limit exists: Step 1: break the integral to two parts: from 0 to 1, from 1 to infinity. Step 2: use limit comparison test for…
Blurred_Vision
  • 93
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Calculate integrals $\int_0^1 {\frac{{\arcsin x}}{x}dx} $

This is my problem,I tried to use change of variable, but no result so far. Can anyone help me? $$\int_0^1 {\frac{{\arcsin x}}{x}dx} $$
Tien Quan
  • 119
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Feynman´s trick to solve $\int_0^\infty \frac{\arctan(x)}{\sqrt{x}(1+x^2)}\,dx$

I wanted to evaluate the integral \begin{align*} \int_0^\infty \frac{\arctan(x)}{\sqrt{x}(1+x^2)}\,dx=\frac{\pi^2}{4\sqrt{2}}-\frac{\pi \ln(2)}{2\sqrt{2}} \tag{1} \end{align*} I thought of using Feynman´s trick by considering the…
Ricardo770
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How to show $\int_0^1 \frac{\ln\frac{1+a}{1+x}}{a-\frac1a{x^2}}dx=\frac{\pi^2}{24}-\frac14\text{Li}_2\left[\left(\frac{a-1}{a+1}\right)^2\right] $

I came across this polylogrithmic integral that evaluates to a close form of dilogarithmic value $$\int_0^1 \frac{\ln\frac{1+a}{1+x}}{a-\frac1a{x^2}}dx=\frac{\pi^2}{24}-\frac14\operatorname{Li}_2\bigg[\bigg(\frac{a-1}{a+1}\bigg)^2\bigg] $$ for…
Quanto
  • 97,352
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Saddle Point Integral

I want to calculate , $$I = \int_0^\infty dx \,x^{2n}e^{-ax^2 -\frac{b}{2}x^4} $$ for real positive a, b and positive integer n. n is the large parameter. Using Saddle Point Integration I find saddle points by setting the derivative P'(x) = 0…
Timtam
  • 227
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$f(0,p)$ for $f(x,p) = \int \frac{\sin(2x)}{x} C_i(x+p) dx$

How do i find $f(0,p)$ for $$f(x,p) = \int \frac{\sin(2x)}{x} C_i(x+p) dx$$ $C_i(x)$ is Cosine Integral. Obviously, if I could solve the integral, I would have substituted $x$ with $0$. But I don't know how to solve it. I wonder if there is a…
Srini
  • 823
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How to integrate $\int{e^{\tan^2{x}}\sin(4x)}dx$

The question says it all. How do I even go about integrating this integral. I'll appreciate some help. Thanks in advance. $$\int{e^{\tan^2{x}}\sin(4x)}dx$$
sayantankhan
  • 2,397
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Integral in $n$-dimensional spherical coordinates

I have to calculate the following integral: $$\frac{c^{-2n}}{\sqrt{2n}}\int_0^\infty\cdots\int_0^\infty \left( \sqrt{x_1 ^2 +\cdots+x_n ^2}\right) (x_1\ldots x_n) \exp{ \left( -\frac{x_1^2+\cdots+x_n^2}{2c^2} \right) } \, dx_1 \cdots dx_n$$ by…
Spine Feast
  • 4,770
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Evaluating $\int_0^\infty\frac{x\tan(ax)}{x^2+b^2}\, \mathrm dx$

The question asks to show that $$\mathcal{I}=\int_0^\infty \dfrac{x\tan(ax)}{x^2+b^2}\mathrm dx=\frac{\pi}{e^{2ab}+1}$$ for $a>0$, $b>0$. I found this on the internet but searched using Approachzero but found no question. I have been trying this…
user730361
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Solve a nasty integral

Can anyone help me with this very nasty integral: $$\int \frac{e^{-\frac{x^2}{2}}~x\left(-1+2b^2+2x^2\right)}{\sqrt{1-e^{x^2}} \sqrt{b^2+x^2}} dx,$$ where $b\in\mathbb{R}$. I've tried pretty much everything I have in bag of tools but nothing worked.…
PML
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