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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Tight(er) lower bound on an integral

I'm trying to evaluate the following integral: $$A(a)=\int\nolimits_{-\infty}^{\infty}e^{-x^2/2}\log\left(\int_{x-a}^{x+a}e^{-t^2/2}dt\right)dx$$ I'll be satisfied with a reasonable lower bound on it. I've tried the Mean Value Theorem bound on the…
M.B.M.
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How to find the integral $\int_{0}^{\infty}\frac{\arctan{x}}{2+x^2}dx$?

Find this integral $$I=\int_{0}^{\infty}\dfrac{\arctan{x}}{2+x^2}dx$$ My try: since let$$\arctan{x}=u$$ then $$I=\int_{0}^{\frac{\pi}{2}}\dfrac{x}{1+\cos^2{x}}dx$$ the wolf <--- then I can't.Thank you
user94270
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On Daniell integral and the notion of measurability

Recently, I decided to learn Daniell integration and after a couple of months on it I like to think that I got the notions right. I also understood the Daniell-Stone theorem that established that, under certain regularity conditions, Lebesgue and…
Maurice
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Find value of integral: $I=\int_0^{2\pi}\frac{dx}{(2+\cos x)^2}$

Find value of integral: $$I_1=\int_0^{2\pi}\frac{dx}{(2+\cos x)^2}$$ and $$I_2=\int_0^{2\pi}\frac{dx}{(2+\sin x)^2}$$ I don't know how, i need a solution, please
Iloveyou
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How to solve this integral: $\int_0^\infty x^a e^{-bx}dx$?

How do i integrate the following definite integral : $\int_0^\infty x^a e^{-bx}dx$. Any hint is appreciated. where $a$ and $b$ are non-negative integers.
Indranil Banerjee
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There exists $c\in [a,b]$ such that $\int_a^c f(t)dt = \int_c^b f(t)dt$

If $f:[a,b]\longrightarrow \mathbf{R}$ is integrable prove that there is $c\in[a,b]$ such that $\int_a^c f(t)dt = \int_c^b f(t)dt$. I set $g(x)=\int_a^x f(t)dt$ but I don't know how I must continue.
user100002
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How find this integral $I=\int\frac{1}{\sin^5{x}+\cos^5{x}}dx$

Question: Find the integral $$I=\int\dfrac{1}{\sin^5{x}+\cos^5{x}}dx$$ my solution: since…
user94270
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integration for |x|<1 for a simple integral

I want to integrate $$\int_{-1}^{1}\frac{dx}{x^3\sqrt{1-x^2}}.$$ I'm not sure where to begin as I have tried integrating by parts but end up in a continuous circle
Lauren
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Integral of derivatives and conjugate

For function $f,g$ in the Schwartz class, I want to show that $$\int_{\mathbb{R}}\left(\frac{d}{dx}f(x)+xf(x)\right)\overline{g(x)}dx=\int_{\mathbb{R}}f(x)\overline{\left(-\frac{d}{dx}g(x)+xg(x)\right)}dx$$ But I don't see how to start, since there…
PJ Miller
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integral of f(x), where f(x) is infinite at one point while zero at other parts

A function f(x) has the value of zero expect one point, where the value is infinite. Does the integral of f(x) equal 0? Or any other values? Thanks a lot.
monkinsane
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Integrals for children

I was just reviewing a chapter on integration defined through step functions, and was wondering how would you explain the concept of an integral of a step function to a child ?
user61913
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Integral $\int_{-\infty}^{\infty}\frac{\mathrm dx}{(ax^2+2bx+c)^{\alpha}}$

let $a>0,ac-b^2>0,\alpha>\dfrac{1}{2}$ show that $$I=\int_{-\infty}^{\infty}\dfrac{\mathrm dx}{(ax^2+2bx+c)^{\alpha}}=\dfrac{(ac-b^2)^{\frac{1}{2}-\alpha}}{a^{1-\alpha}}\dfrac{\Gamma{(\alpha-\dfrac{1}{2})}}{\Gamma{(\alpha)}}\sqrt{\pi}$$ This problem…
user94270
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How find this integral $I=\int_{0}^{\infty}\frac{\log{\cos^2{x}}}{1+e^{2x}}dx$

find the value $$I=\int_{0}^{\infty}\dfrac{\log{\cos^2{x}}}{1+e^{2x}}dx$$ My try: let $$e^{2x}=u\Longrightarrow x=\dfrac{1}{2}\log{u}$$ then $$I=\int_{1}^{\infty}\dfrac{\log{(\cos^2{(\dfrac{1}{2}\log{u})})}}{2u(1+u)}du$$ then I can't.Thank you
math110
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Question about indefinite integral

Let's say I have expression with multiplication which has variable x $$\int x^2e^{x^3}dx$$ So in example it shows $$\int x^2e^{x^3}dx=\frac{1}{3}\int e^u du=\frac{1}{3}e^u+C=\frac{e^{x^3}}{3}+C$$ $u=x^3$ $du=3x^2 dx$ So I don't understand from…
Templar
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2 answers

Evaluating $\int{\frac{1}{\sqrt{x^2+y^2}}\mathrm dx}$

Attempting to calculate $\displaystyle \int{\dfrac{1}{\sqrt{x^2+y^2}}\mathrm dx}$, $$\int{\dfrac{1}{\sqrt{x^2+y^2}}\mathrm dx}=\int{\frac{1}{\sqrt{(y\tan\theta)^2+y^2}}y\sec^2\theta \mathrm d\theta}=\int{\sec\theta d\theta}=\ln(\sec\theta…
resgh
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