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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Evaluate the integral $\int_{0}^{+\infty}\left(\frac{x^2}{e^x-1}\right)^2dx$

Find the value of the integral $$\int\limits_{0}^{+\infty}\left(\frac{x^2}{e^x-1}\right)^2\;\mathrm{d}x\;.$$ We can let $x=\ln{t}$ to get $$\int\limits_{1}^{+\infty}\frac{(\ln{t})^4}{t(t-1)^2}\;\mathrm{d}t\;.$$ But how can we evaluate it from…
math110
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Looking for a closed form of $I(n,m)=\int_0^{+\infty} e^{-ax^n-\frac{b}{x^m}} \, dx$

I am looking for a closed form of this integral for reals $a,b>0$ and integers $n,m>0$ $I(n,m)=\int_0^{+\infty} e^{-ax^n-\frac{b}{x^m}}$ I read about $I(2,2)$ in the book Irresistible integrals. However, the neat proof included in the book does not…
Evariste
  • 2,511
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What is running integral?

My question might be too simple. But I could not find any source giving the answer. Can you please explain the running integral?
user1067742
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Is every monomial over the UNIT OPEN BALL bounded by its L^{2} norm?

Let $m\geq 2$ and $B^{m}\subset \mathbb{R}^{m}$ be the unit OPEN ball . For any fixed multi-index $\alpha\in\mathbb{N}^{m}$ with $|\alpha|=n$ large and $x\in B^{m}$ $$|x^{\alpha}|^{2}\leq \int_{B^{m}}|y^{\alpha}|^{2}dy\,??$$
GoldSoundz
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If $f:[0,1] \rightarrow \mathbb{R},f(0)=0$,is convex and integrable,prove that:$\int_{0}^{1}f(x)dx\ge(2n+1)\int_{0}^{1}(1-x^{\frac{1}{n}})f(x)dx$.

If $f:[0,1] \rightarrow \mathbb{R},f(0)=0$,is convex and integrable,prove that:$\int_{0}^{1}f(x)dx\ge(2n+1)\int_{0}^{1}(1-x^{\frac{1}{n}})f(x)dx$. My progress: after simplifying I got $2n \int_{0}^{1}f(x) dx \le (2n+1) \int_{0}^{1}…
user321656
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Show that $\pi=\frac{22}{7}-\frac{\pi}{4}\int_0^{\infty}\frac{e^{-2x\pi}\left( 1-e^{-\frac{x\pi}{2} } \right)^4}{\cosh\left(\frac{x\pi}{2}\right)}dx$

An integral from maths world; pi formulas (50), $$\pi=\frac{22}{7}-\int_0^1\frac{x^4(1-x)^4}{1+x^2}dx$$ We found another similar integral to it, via experimental with wolfram integrator,…
user334593
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Example of $f:]-1,1[ \rightarrow \mathbb{R}$ such that $G(x)=\int_0^x f(t)dt$ is not $f$'s primitive?

I cannot figure out the following: Give an example of integrable $f:]-1,1[ \rightarrow \mathbb{R}$ such that $G(x)=\int_0^x f(t)dt$ is not $f$'s primitive?
mavavilj
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A certain multidimensional integral.

Consider a following multidimensional integral: \begin{equation} \bar{I}^{(t_0,t)}_p := \int\limits_{t_0 \le \xi_0 \le \cdots \le \xi_{p-1} \le t} \prod\limits_{j=0}^p (\xi_{j-1}-\xi_j) \cdot \prod\limits_{j=0}^{p-1} \frac{d…
Przemo
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Symbolic integration of a polynomial with large exponents

Motivation/background: The following integrals were part of an argument in my PhD thesis that a certain piece of forensic evidence has no inculpatory value. I gave up trying to solve them analytically using Mathematica, and settled instead for…
Dustin Wehr
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How to integrate $\frac{1}{x\sqrt{x}}$

How to integrate $$\frac{1}{x\sqrt{x}}$$ I don't see how could I use u substitution or integration by parts. I tried both, but it just got worse(more complex). I haven't integrate in years and I just can't warp my head around this. Edit: Thank you…
Szőke Szabolcs
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definition of integration using equally spaced intervals

if $f$ is Riemann integrable then $\exists$ sequence $D_{n}$ of dissections such that $S_{D_{n}} - s_{D_{n}}$ tends to $0$ is the above equivalent to saying: consider ${D_{n}} = a + \frac{k(b-a)}{n} $ interval is $[a,b]$ $0 \leq k \leq n$ $f$ is…
zebra1729
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What is the difference between ($\tan x \sec^2x$) and ($\sin x/\cos^3x$)? Why is the answer to the integration different?

$$\int \:\frac{\left(\sin x+\tan x\right)}{3\cos^2x}dx$$ I know I have to split the equation into $$\frac{1}{3}\int \:\left(\:\frac{\sin x}{\cos x}\right)\left(\frac{1}{\cos x}\right)dx+\frac{1}{3}\int \:\left(\:\tan…
Feefee
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Complicated Indefinite Integration

$\displaystyle\int{\dfrac{\left(\sqrt{x}+2\right)^2}{3\sqrt{x}}\;\mathrm{d}x} =\frac{2}{3}\big(\sqrt{x}+2\big) $ I do not know how to get $\dfrac{2}{3}$. I assume $\sqrt{x}+2 = u$, then $\mathrm{d}u= \dfrac{1}{2}\,x^{1/2}$ But it is wrong…
user319316
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convergence of integral for s>n/2

I want to prove that $\int_{\mathbb{R^n}}\frac{1}{(1+|x|^2)^s}dx$ is convergent for $s>n/2$. The only thing which comes to my head is using the $\arctan$ function, but then I got stuck. Any hints are welcome! Thank you!
Kerr
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Integral formula for surface area of revolution

To find volume of the shape generated by rotating $f(x)$ around the $x$ axis we calculate $$\int_a^b \pi f(x)^2\, \mathrm d x$$ as the area of a circle is $\pi r^2$ and we just split it into discs. Why then by analogy is the surface area formula not…
user85798
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