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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Asymptotic expansion of integral depending on a parameter

I would like to find an equivalent (first term of an asymptotic expansion) of the following function in $x=0$ and $x=1$: $$f(x)=\int_0^{+\infty} \frac{1}{t^x\sqrt{1+t^2}}\ dt$$ My professor gave a highly unmotivated solution which doesn't invoke the…
math_lover
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Another arctan integral

$$\int_{-1}^{1}\arctan(zx){\mathrm dx\over x\sqrt{1-x^2}}=\pi\sinh^{-1}(z)\tag1$$ $x=\sin{y}$ then $dx=\cos{y}dy$ $$\int\arctan(z\sin{y}){\mathrm dy\over \sin{y}}\tag2$$ $$\sum_{k=0}^{\infty}{(-1)^k\over z^{2k+1}}\int{\mathrm dy\over…
gymbvghjkgkjkhgfkl
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Involving Euler's constant and Gamma function

$$\int_{0}^{\infty}{e^{-x}+x-1\over x(e^{x/a}-1)}\mathrm dx=a\gamma+\ln{\Gamma(1+a)}\tag1$$ $\gamma$ is Euler-Mascheroni constant, How can we show that $(1)=a\gamma+\ln{\Gamma(1+a)}?$
gymbvghjkgkjkhgfkl
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Improper integral: $\int^{+\infty}_0e^{-(a^2t^2+b^2/t^2)}$

Consider the application $f: (\mathbb{R}^{+*})^2 \to \mathbb{R}$ given by $f(a,b)=\int^{+\infty}_0e^{-(a^2t^2+b^2/t^2)}$ Calculate $f(a,b)$. I thought to take the derivative inside the integral sign (this needs justification) I…
math_lover
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Show that $\int_{0}^{\pi}{\sin^5(x)\over 1+\cos^3(x)}\mathrm dx =\ln{3}?$

$$\int_{0}^{\pi}{\sin^5(x)\over 1+\cos^3(x)}\mathrm dx =\ln{3}\tag1$$ How can we show that $(1)=\ln{3}$ $u=\sin^3{x}$ then $du=3\sin^2{x}\cos{x}dx$ $${1\over 3}\int{u\mathrm du\over \cos{x}+\cos^4{x}}\tag2$$ This is not a good substitution
gymbvghjkgkjkhgfkl
  • 8,090
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Prove $∫e^x \, dx=e-1$ using rectangular method

I need to prove the following using rectangular method: $$∫_0^1e^x\,dx=e-1.$$ Otherwise speaking, I need to solve integral via area of ​​curvilinear trapezoid. And I have no idea how to do that. Would really appreciate your help.
Nika J
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evaluate cosine power reduction formula in terms of Gamma function

I ran across an identity I had not saw before, and am wondering how it can be derived. $\displaystyle \int_{0}^{\frac{\pi}{2}}\cos^{m}(x)\cos(nx)dx=\frac{\pi\Gamma(m+1)}{2^{m+1}\Gamma(\frac{m+n}{2}+1)\Gamma(\frac{m-n}{2}+1)}$. For the case, $m=n$,…
Cody
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Calculation of $\int \sin^2 x\frac{\sin x-\cos x}{\sin x+\cos x}dx$

Calculation of $\displaystyle \int\frac{\sin^2 x\cos x}{ \sin x+\cos x}dx$ and $\displaystyle \int^{\pi}_{0}\frac{1}{1-2a\cos x+a^2}dx, ,0
DXT
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Integral of dependent variables

For the computational sake of integration, how would we integrate something such as $\int x^2z^2 \space dz$ if we had that $z^2=x^2+y^2$ and that $0 \le z \le 1$ were equations defining a cone in $\mathbb R^3$, where $x$ and $y$ are any points…
user258521
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Evaluate $2\int_{0}^{\pi\over 2}\ln^2(\tan^2{x})\mathrm dx$

How to show that $(1)$ $$2\int_{0}^{\pi\over 2}\ln^2(\tan^2{x})\mathrm dx=\pi^3?\tag1$$ $u=\tan^2{x}$ then $\mathrm dx={\mathrm du\over \sqrt{u}(u-1)}$ $$\int_{0}^{\infty}\ln^2{u}{\mathrm du\over…
gymbvghjkgkjkhgfkl
  • 8,090
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When does something "go to zero fast enough"?

While reading answers here, an argument is often that a term in an integral does not go to zero fast enough, resulting in $\infty$. Some examples: 1, 2, 3, ... Is there an intuitive explanation for why the speed of going to zero matters; Is there a…
Frans Rodenburg
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Evaluate $\int_{0}^{\pi\over 4}{\ln(x_n)\over \cos^2(x)}\mathrm dx$

Let $$x_1={1 \over \tan(x)}$$ $$x_2={1+\tan^2(x)\over 2\tan(x)}$$ $$x_3={1+3\tan^2(x)\over 3\tan(x)+\tan^3(x)}$$ $$x_4={1+6\tan^2(x)+\tan^4(x)\over 4\tan(x)+4\tan^3(x)})$$ $$x_n={{n\choose 0}+{n\choose 2}t^2+{n\choose 4}t^4+\cdots\over {n\choose…
gymbvghjkgkjkhgfkl
  • 8,090
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$\int \frac{1}{x^2\sqrt{x^2+x+1}}\, dx$

How would one go about integrating $\int \frac{1}{x^2\sqrt{x^2+x+1}}\, dx$ I tried rationalizing and then doing partial fractions but got something really ugly and IBP doesn't work too well either.
WhatsDUI
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Intuition for the integral version of the chain rule

$$\int f(g(x))\,g'(x)\,dx=\int f(u)\,du$$ But... why? I know I can take $u=g(x)$ so $du=g'(x)\,dx$. I know how to apply the rules but I got lost on the intuition of what the rules actually do and why they work. So the question is really: why,…
GambitSquared
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Prove that $\int_a^{a+T}f(x)dx$ is independent of $a$ if $f$ is continuous and periodic with period $T$

Prove that $\int_a^{a+T}f(x)dx$ is independent of $a$ if $f$ is continuous and periodic with period $T$ I indeed don't how to treat to this problem.
Nancy Rutkowskie
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