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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area of a triangle using double integral

I can find the area of a triangle with known vertices but the problem here is that the question is general: I have to use double integral to prove that the area of the triangle is: $$A_{\text{triangle}}=\frac {\text{base}\times\text{height}}{2}$$ I…
mhmd
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Order limit vs Integral

I'm thinking about when can be stated that $\lim\limits_{s\to s_0}\displaystyle\int f(x,s)dx = \int \lim\limits_{s\to s_0}f(x,s)dx$ Can you help me with this? What are the hypothesis about $f$ to assure the statement? Particularly I was trying to…
MathGuest
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Integral Equation $\int_{-\infty}^\infty \frac{f(y)}{1+(x-y)^2}\mathrm dy =0 \quad \forall x$

I want to calculate $f(y)$ such that $$\int_{-\infty}^\infty \frac{f(y)}{1+(x-y)^2}\mathrm dy =0 \quad \forall x$$ Can we prove that the solution to this problem is $f(y)=0?$ Thank you so much
Mamal
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Compute a $2$D integral

$$I=\iint_{D}\cfrac{dxdy}{\sqrt{1-\cfrac{{x}^{2}}{{a}^{2}}-\cfrac{{y}^{2}}{{b}^{2}}}\times\left( {x}^{2}+{y}^{2}+1-\cfrac{{x}^{2}}{{a}^{2}}-\cfrac{{y}^{2}}{{b}^{2}}\right)^{\cfrac{3}{2}}}$$ where…
XLDD
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Another Basic Integration Question, Possibly By Substitution

What's the integral of $f(x)=(1-x^2)^{1/2}$? I tried making $x=\sin(t)$ and doing integration by substitution but I don't think I arrived to the correct answer. All responses are appreciated...
Vladimir Nabokov
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Show that $\sin(x^2)$ is integrable around $\infty$.

I have to show that $f(x)=\sin(x^2)$ is integrable on $[1, \infty[$. This is French terminology, so "intégrable" specifically means that the integral of $|f|$ exists. The only method I know is to compare it to functions of the form…
Jack M
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How am I supposed to substitute these integral bounds?

I want to solve the integral $$\int\nolimits_0^1 \int\nolimits_0^{1-x} \exp \left(\frac{y}{x+y} \right) \; \mathrm dy \; \mathrm dx$$ and I am given the hint to use the substitution $x+y=u$ and $y = uv$. Now, I've never done such multiple…
Huy
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System of equations with integration

I'm having trouble trying to solve the following system. $f(x) = 1 + \int_0^x{g(t)\mathrm{d}t}$ $g(x) = x(x - 1) + \int_{-1}^{1}{f(t)}\mathrm{d}t$ I have in mind to substitute this integrations for constants, but I'm kind lost. Can someone help…
aajjbb
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How to find $\int_{0}^{\infty}\frac{\tan^2{x}}{x^2}\,dx$?

Find the integral $$\int_{0}^{\infty}\frac{\tan^2{x}}{x^2}\,dx=? \tag{1}$$ and $$\int_{0}^{\infty}\frac{x^2}{\tan^2{x}}\,dx=?\tag{2}$$ I know…
math110
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Evaluate $\int _0 ^1 \frac{e^{ax}-e^{bx}}{(e^{ax}+1)(e^{bx}+1)} \mathrm{d}x$

Evaluate $$\int _0 ^1 \frac{e^{ax}-e^{bx}}{(e^{ax}+1)(e^{bx}+1)} \mathrm{d}x$$ where $a$ and $b$ are constant. I attempt to factor the fraction but then have no ideas where to go. Please help me. Thanks!
user1500178
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integration by parts

!!!! PD: I did a little change in the denominator !!!! I need to solve this integral using integration by parts. $\displaystyle\int\frac{x\,dx}{\sqrt{(a^2+b^2)+(x-c)^2}}$ Thanks! PS: I know that I can to…
yemino
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Strange delta function

I don't know what to do when I see a delta-function of the following sort appear in an integral (3d-spherical here): $$\delta^3(r\sin \theta - r_0).$$ E.g. the argument is a function of two of the variables. I'm familiar with the standard properties…
rj7k8
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Can you integrate a quadratic raised to an arbitrary integer power?

I would like to integrate: $\int_0^\infty (ax^2 + bx + c)^n dx$, where $n$ is a negative integer. Does anyone know a method to do this? My obvious first try has been using Mathematica, but it doesn't find a solution. And, I've not found it by going…
Matt Pitkin
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Simplify nested, dependent integral

I am trying to simplify an integral that appeared in a problem I'm working on, $$\int_{s_1=0}^1\int_{s_2=s_1}^1\int_{s_3=s_2}^1\cdots \int_{s_n=s_{n-1}}^1\prod_{i=1}^n f(s_i) ds_n ds_{n-1} \cdots ds_1.$$ Here $n$ is a positive integer and $f(s)$ is…
imladris
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Prove $\int_0^1 (\sqrt[n]{1-x^n}-x)^2 dx=\frac{1}{3}$ for $n \gt 0$

I have found that the value of the integral below is always $1/3$ for all positive $n$. $$\int_0^1 (\sqrt[n]{1-x^n}-x)^2 dx$$ Can anyone prove this for me? Thank you.
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