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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Integral: $\displaystyle\int_0^\infty\!\frac{\tanh^2 x}{x^2}\,dx$

In this answer, Mr. Noam Shalev - nospoon showed that $$ I = \int_0^\infty \! \frac{\tanh^2 x}{x^2} \, dx = \frac4{\pi^2} \int_0^1 \! \frac1x \, \log^2 \frac{1-x}{1+x} \, dx $$ with using the following substitution $$ x = \frac{1-t}{1+t} ,\quad dx =…
Kei Tojo
  • 404
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Computing $\iint x^2 \tan(x) + y^3+ 4 dx dy$ in a circle.

I have the following double integral and the following domain. $$\iint x^2 \tan(x) + y^3+ 4 dx dy$$ $$D=\{(x,y): x^2+y^2\le2\}$$ I know the domain is a circle and it can be described as: $$D=\left\{(x,y): -\sqrt{2} \le x \le \sqrt{2} \text{ and }…
user3347814
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About an exotic integral

I would like to show that : $$\displaystyle\int_0^\infty \frac{x^{\frac\pi5-1}}{1+x^{2\pi}} \mathrm dx=\phi$$ ($\phi$ is golden ratio) Apparently it comes from a more general form : $$\int_0^\infty \frac{x^{\pi/k-1}}{1+x^{2\pi}}\mathrm…
LexLarn
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How do you evaluate $\int \frac{y^2}{y^2+d^2}dy$?

$\color{green}{question}$: How do you evaluate this integral? $$\int \frac{y^2}{y^2+d^2}dy=y-d\,{\tan}^{-1}\left ( \frac{y}{d} \right )+\mathrm{constant}$$ $\color{green}{I~know}$ I should use the change of variables, But I do not know how to…
Software
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General form of a double integral

I have difficulty calculating the general form of a double integral: $\int_0^1\int_0^1\sqrt{x^2+y^2-2mxy}\,\text{d}x\text{d}y$, where m is a constant between -1 and 1. I've tried some special…
Aaron Lee
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Difficult Definite Integral with inverse Cosh

How can I solve this integral containing inverse cosh? Does it have any antiderivative? $$ \int_b^r t^2 \operatorname{arccosh}(a/t) \sqrt{r^2 - t^2} d t$$ for $0< b< r< a$.
Billo
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Trapezoidal Numerical Integration formula variations

I am reading a book (SECOND EDITION) on computational physics by N.J. Giordano and H. Nakanishi. On Appendix E, page 502, the trapezoidal rule is introduced as: $$\int_a ^b f(x)dx \approx \sum_{i=1} ^{N-1} f(x_i) +\frac{1}{2}[f(a)+f(b)]$$ Why is…
RMS
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Find $\omega$ such that $\infty>\int^1_0r^{2n+1}\omega(r)dr\ge n$

I am looking for a $\omega $ such that for any $n\in \mathbb{N}_0$ $$\infty>\int^1_0r^{2n+1}\omega(r)dr \ge n$$ Well $\omega \notin L^{p}([0,1]$ due to hoelder inequality $$\int^1_0r^{2n+1}\omega(r)dr \le \left(…
CoffeeArabica
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Integrating a faction with $e^x$ in the denominator

Let $I$ be the integral we are asked to evaluate. We have: $$ I = \int \dfrac{1}{e^x \left( e^{x}+1\right) } \,\, dx $$ Now, I apply the technique of partial fractions. \begin{align*} \dfrac{1}{e^x \left( e^{x}+1\right)} &= \dfrac{ A }{e^x+1} +…
Bob
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Show that $\frac{1}{(k+1)^p}\leq \int_k^{k+1}\frac{1}{x^p}dx\leq\frac{1}{k^p} $ for every positive $k$ and $p\geq1$

Let $p\geq1$. Show that $$\frac{1}{(k+1)^p}\leq \int_k^{k+1}\frac{1}{x^p}dx\leq\frac{1}{k^p} $$ for every positive $k$. Hence show that $$\sum_{k=2}^{n+1}\frac{1}{k^p}\leq\int_1^{n+1}\frac{1}{x^p}dx\leq\sum_{k=1}^n\frac{1}{k^p}$$ for every positive…
elizabeth10
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How to find integral : $\int\;\frac {x^2+3}{x^6(1+x^2)}dx$

Question: How to find integral : $$\int\;\frac {x^2+3}{x^6(1+x^2)}dx$$ My attempt: \begin{align} \int\;\frac {x^2+3}{x^6(1+x^2)}dx＝\int\;\frac {x^2＋1＋2}{x^6(1＋x^2)}dx & ＝\int\;\frac {x^2＋1}{x^6(1＋x^2)}＋\frac {2}{x^6(1＋x^2)}dx \\ &＝\int\;\frac…
Moustapha_M_I
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show that $\int_0^\infty x^n\,\text{sech}^2x\,\mathrm dx=(2^{1-n}-4^{1-n})\,\Gamma(n+1)\,\zeta(n)$

show that $$\int_0^\infty x^n\,\text{sech}^2x\,\mathrm dx=(2^{1-n}-4^{1-n})\,\Gamma(n+1)\,\zeta(n)$$
mnsh
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How to solve $\int_{0}^{1}e^{x}\ln(1+e^{x})dx$?

I am trying to solve this integral: $\int_{0}^{1}e^{x}\ln(1+e^{x})dx$ Here is my attempt: $[e^{x}\ln(1+e^{x})]_{0}^{1} - \int e^{x} \frac{e^{x}}{1+e^{x}}dx$ $[e^{x}\ln(1+e^{x})]_{0}^{1} - \int e^{x} \frac{e^{x}}{1+e^{x}}dx$…
Firellsp
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Can I get from $\frac{d^2 x}{dt^2} = g$ to $\iint d^2 x = \iint g\,dt^2$?

The second law of Newton is $$a = \frac{F}{m}$$ and so in a constant gravitational field with gravity $g$ I have $$a = \frac{d^2x}{dt^2} = g$$ To solve this I multiply with $dt^2$ and get $$d^2x=g\,dt^2$$ and integrate $$\iint d^2x = g\,\iint…
Alex
  • 132
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Evaluate $\underbrace{\idotsint}_n \exp\left(-\sum_{1\le i\le j\le n}^n x_i x_j\right) \mathrm{d}x_1\cdots\mathrm{d}x_n $

I am currently studying V.I. Arnold's course, and I am stuck on this exercise: Evaluate $$ \underbrace{\idotsint}_{n} \exp\left(-\sum_{1\le i\le j\le n}^n x_i x_j\right) \mathrm{d}x_1\cdots\mathrm{d}x_n $$ Can anyone suggest a hint? Thank you in…
zytsang
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