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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Integral check. Is partial fractions the only way?

I'm getting a different answer from wolfram and I have no idea where. I have to integrate: $$\int_0^1 \frac{xdx}{(2x+1)^3}$$ Is partial fractions the only way? So evaluating the fraction first: $$\frac{x}{(2x+1)^3} = \frac{A}{2x+1} +…
Jwan622
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Evaluating $\int_{} \frac{xe^{2x}}{(1+2x)^2}dx$ via integration by parts

$\int_{} \frac{xe^{2x}}{(1+2x)^2}dx$ I am having trouble picking the correct $u/dv$ before integrating by parts. I felt like L.I.A.T.E. did not really help me here... This is what I tried, but it ended up with integration spiraling into an endless…
Evan Kim
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$\int_{-\infty}^{\infty} \frac{\mathrm dx}{\cosh(x)^n}$: does it work?

I am trying to evaluate $$J(n)=\int_{-\infty}^{\infty} \frac{\mathrm dx}{\cosh(x)^n}=2\int_{0}^{\infty} \frac{\mathrm dx}{\cosh(x)^n}$$ for $n\in\Bbb N$. I started with $t=\tanh\frac{x}2$: $$J(n)=4\int_0^1\frac{(1-t^2)^{n-1}}{(1+t^2)^n}\mathrm…
clathratus
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$\int_{0}^{1} t^2 \sqrt{(1+4t^2)}dt$ solve

$$\int_{0}^{1} t^2 \sqrt{(1+4t^2)}dt$$ my attempt $$t = \frac{1}{2}\tan(u)$$ $$dt = \frac{1}{2}\sec^2(u)du\\$$ $$\begin{align} \int_{0}^{1} t^2 \sqrt{(1+4t^2)}dt&=\int_{0}^{1} \frac{\tan^2(u)}{4}…
Tinler
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On $F(n;a)=\int_0^\pi\frac{\mathrm dx}{(1+a\cos x)^n}$

I am currently working on, out of interest, the integral $$F(n;a)=\int_0^\pi \frac{\mathrm dx}{(1+a\cos x)^n}$$ For $a\in(-1,1)$ and $n\in \Bbb N$. I would like to know if my methods are correct or if there are any other ways to go about this.…
clathratus
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How to integrate $\int\frac{x}{(x^2-2x+5)^2} \, dx$?

$$\int\frac{x}{(x^2-2x+5)^2} \, dx$$ I tried to complete the square of the bottom like this $\int\frac{x}{((x-1)^2+4)^2} \, dx$ but I'm still not sure what to do. Any help would be appreciated. Thanks
Pie Man
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The closed form representations of Integrals of logarithm functions

I wish to find a closed form representations of the following integral $$\int\limits_{0}^1\frac{\log^p(x)\log^r\left(\frac{1-x}{1+x}\right)}{x}dx=?$$ Here $p\ge 1$ and $r\ge 0$ are nonnegative integers. It can be expressed in terms of a linear…
xuce1234
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What's the intuition behind this chain rule usage in the fundamental theorem of calc?

I understand this: but I don't understand this example fully: So I get the intuition behind the idea that the sliver at some point x in the area function (g(x)) is just the y coordinate of the original function (f(x)), but I'm sure why we need…
Jwan622
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Change of variables theorem problem

I'm having trouble with the change of variables theorem in two variables. The theorem says: $$\iint f(x,y)dxdy=\iint f(x(u,v),y(u,v))|J|dudv$$ Where J is the Jacobian. If $f(x,y)=xy$ where $x,y \in \mathbb{R} $ $$\iint f(x,y)dxdy=…
Jorge
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Infinite Integration using the ceiling function

$$\int_0^\infty \frac{\sin(x\pi)}{\lceil x \rceil^2 + \lceil x \rceil} dx$$ My teacher recently gave me this and it's stumped me.
user546944
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Need help calculating $\int_{0}^{\infty} \frac{1- \cos (t)}{t^\alpha} {\rm d}t$

As part of an larger assignment I need to calculate the following integral $$ \int_{-\infty}^{\infty} \frac{1-\cos(\lambda x)}{|\lambda|^\alpha} {\rm d}\lambda \quad x \in \mathbb{R}, \,1 < \alpha \leq 2 $$ I substituted $t=\lambda|x|$ to…
brodz
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An Awkward Integral Arising in Scattering of $\alpha$-Particle by Nucleus

I wonder whether anyone knows how to do the integral $$\int\frac{dr}{r\sqrt{r^2 -br\exp(-kr)-a^2}} .$$ It arises when the Rutherford scattering problem is instisted upon being treated classically instead of by treating the incoming…
AmbretteOrrisey
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Feynman's Integration technique, parameter finding

Recently I was studing the Feynman integration technique (differentiation under the integral sign), but I allways get stuck on the general function definition, and I was wondering if there is some kind of formula, trick, table, or something else…
liuzp
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Triple Integrate $x^{2n}+y^{2n}+z^{2n}$

Evaluate: $$\iiint x^{2n}+y^{2n}+z^{2n} \; \mathrm dV$$ over the region $x^2+y^2+z^2 \le 1$ I think I know how to find the limits, but I don't know how to do the integration. I see that I have to do a change of variable, but I don't know what…
dessskris
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A general expression for $\int_{-\infty}^{\infty}\frac{f(x)}{1+x^2}dx$ in terms of the series expansion coefficients of $f(x)$

An integral of the form $$\int_{-\infty}^{\infty}\frac{f(x)}{1+x^2}dx\tag{1}$$ can sometimes be easily computed if $f(x)$ can be analytically continued to the complex plane using contour integration. E.g. if $f(z)$ is an entire function that tends…
Count Iblis
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