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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Find volume of the cone using integration

A cone can be though as a concentration of circles of radius tending to $0$ to radius $r$ and there will be infinitely many such circles within a height of $h$ units. Area of one such circle of radius $r$ will be $\pi r^2$. Volume of cone = sum of…
Mathejunior
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How to integrate by using imaginary and real part of $e^{ix}$

I want to calculate integrals with $\sin(x)$ and $\cos(x)$ by real and imaginary part of $e^{ix}$. Assume $e^{ix}=\cos(x)+i\sin(x)$ For example $$ \begin{align*} \int \sin(x)dx &= \int Im(e^{ix})dx = Im\left(\int e^{ix}dx\right)=…
boucekv
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Evaluate $\int {2x\over x^2-1}dx$

My friend evaluated this before he went to bed: $$\int {2x\over x^2-1}dx$$ The answer was $\log(x^2-1)$. I just can't figure out how that works. I know that $\int \frac1x dx = \log|x|$, so what just happened to $2x$?
Kishan Thobhani
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Evaluate $\lim_{r\to\infty}\frac{\int_0^{\pi/2}x^{r-1}\cos x\,\mathrm dx}{\int_0^{\pi/2}x^r\cos x\,\mathrm dx}$

How I can evaluate $$\lim_{r\to\infty}\frac{\int_0^{\pi/2}x^{r-1}\cos x\,\mathrm dx}{\int_0^{\pi/2}x^r\cos x\,\mathrm dx}$$ I have tried by replacing $x$ with $y\pi/2$ then the limits would change from zero to $1$ but the integrals would cancel as…
Varun Maddipati
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Integral involving $\phi$

$$\int_{0}^{\pi/2}\arctan\left({2\over \cos^2{x}}\right)\mathrm dx=\pi\arctan\left({1\over \sqrt{\phi}}\right)\tag1$$ $\phi$ is the golden ratio $2\sec^2{x}=2\tan^2{x}+2$ $u=\sec^2{x}$ then $\mathrm du=2{\tan{x}\over \cos^2{x}}\mathrm dx$…
gymbvghjkgkjkhgfkl
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Nested integral: volume of a right simplex

Consider the integral $$I_n \equiv \int_{t_1=0}^T dt_1 \int_{t_2 = t_1}^T dt_2 \cdots \int_{t_n = t_{n-1}}^T dt_n \, .$$ I suspect the result should be $I_n = T^n/n!$ but would like to prove it. Doing the $n^\text{th}$ integral…
DanielSank
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Concerning this integral $\int_{0}^{1}\left({1\over \ln(x)}+{1\over 1-x} -{x^s\over 2}\right){\mathrm dx\over 1-x}$

Motivation from this question $(1)$ $$\int_{0}^{1}\left({1\over \ln(x)}+{1\over 1-x} -{1\over 2}\right){x^s\over 1-x}\mathrm dx=F(s)\tag1$$ setting $s=0$ then $F(0)=-{1\over 2}+{1\over 2}\ln(2\pi)-{1\over 2}\gamma$ by a slight variation of $(1)$, we…
gymbvghjkgkjkhgfkl
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How to compute $\int_0^\infty e^{-a(s^2+1/s^2)}\, ds$

How do I integrate $$\int_0^\infty e^{-a(\frac{1}{s^2} + s^2)}\, ds \tag{*}$$ Context: At page 602 of the paper "Reaction-Diffusion equations for Interacting Particle Systems" from De Masi Ferrari and Lebowitz one…
Conrado Costa
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Riemann integrable and continuous almost everywhere.

I want a proof for this theorem: Let $f$ be a function on $[a,b]$. Then $f$ is Riemann integrable if and only if $f$ is bounded and continuous almost everywhere.
mshj
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An integral involving error functions and a Gaussian

Let $d\ge 1$ be an integer and let $\vec{A}:=\left\{ A_i \right\}_{i=1}^d$ be real numbers. We consider a following integral: \begin{equation} {\mathfrak I}^{(d)}(\vec{A}):=\int\limits_0^\infty e^{-u^2}\left[ \prod_{i=1}^d \operatorname{erf}(A_i u)…
Przemo
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Compute integral $\int_1^3\frac{\ln x}{x^2+3}\ dx $

How to solve the integral $$\int_1^3\dfrac{\ln x}{x^2+3}\ dx\ ?$$ I was thinking of substituting $t=\ln x$ and then factor one $x$ out,is it correct?
Lola
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Computing $ I_{n}=\int \tan(x)^n \mathrm dx$

I'm trying to compute: $$ I_{n}=\int \tan(x)^n \mathrm dx$$ We have: $$ I_{n}+I_{n-2}=\int (1+\tan(x)^2)\tan(x)^{n-2} \mathrm dx$$ $$ I_{n}=\frac{1}{n-1}\tan(x)^{n-1}-I_{n-2}+C$$ Which gives the formulas: $$ \int \tan(x)^{2n} \mathrm dx=…
Chon
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Proving: $\operatorname{P.V.} \int^\infty_{-\infty}\frac{\ln(t^2+1)}{t^2-1}dt =\frac{\pi^2}{2}$

How to prove that $$\operatorname{P.V.}\int^\infty_{-\infty}\frac{\ln(t^2+1)}{t^2-1}dt =\frac{\pi^2}{2}\: ?$$
min
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Using Feymann's trick in Advanced Integration

Introduction: $\def\d{\mathrm{d}}$A common integration technique is to employ Feymann's trick. Assume that we have the following function of two variables$$\int_a^bf(x,y)\, \d x$$Then we can differentiate with respect to $y$ provided that $f$ is…
Crescendo
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Calculate $\lim _{n\to \infty }a_n\int _0^1 x^{2n}\sin \frac{\pi x}{2}dx$

I have to calculate $$\lim _{n\to \infty }a_n\int _0^1 x^{2n}\sin \frac{\pi x}{2}dx$$ Where $$a_n = \sum _{k=1}^n\sin \frac{k\pi }{2n}$$ I have found that $$\lim _{n\to \infty} \frac{a_n}{n} = \frac{2}{\pi} $$ if that helps in any way.
Liviu
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