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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How to integrate $\int e^{-x}\sin(3x)\;dx$?

I want to integrate following by the method of integration by parts $$\frac{\cos(3x)}{e^{x}}$$ when I try to solve it by integration by parts it always leads to something like as mentioned below and it still have integration sign around it.…
Labeeb
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How can you integrate $\int \exp{\left( -x-e^{-x}\right)}dx$

How can you integrate $$\int \exp{\left( -x-e^{-x}\right)}dx$$ I have tried some substitutions but they make things harder than before. Is there some trick that I can use to solve this? Finishing the integration based on the anwers: Let $u=-e^{-x}…
Nullspace
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Integral over the Japanese bracket

I want to show that $$\int \langle x\rangle^{-1-\epsilon} dx = \int (1+|x|^2)^{\frac{-1-\epsilon}{2}}dx$$ converges for $\epsilon>0$. Assume we're on $\mathbb{R}$. Because of symmetry we can integrate from $0$ to $\infty$ modulo a…
Jakob Elias
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For a continuous function $f (t ), 0 ≤ t ≤1,$ the integral equation...

I am stuck with the following problem: For a continuous function $f (t ), 0 ≤ t ≤1,$ the integral equation $y(t)=f(t)+3 \displaystyle \int_{0}^{1}tsy(s)ds \,$ has (a) a unique solution if $\displaystyle \int_{0}^{1}sf(s)ds \ne 0$ (b) no…
learner
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Integration by substitution problem $x = C \sin(t)$.

For solving the integral: $$ \int_a^b \sqrt{\alpha^2 - \beta^2 x^2} \, dx $$ I've been taught to use $x = \frac{\alpha}{\beta} \sin(t)$ in order to get $$ \frac{\alpha^2}{\beta} \int_{\arcsin(a \beta/\alpha)}^{\arcsin(b \beta/\alpha)}…
Victor Moore
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irrational integral $ \int \frac{1+\sqrt{x^2+3x}}{2-\sqrt{x^2+3x}}\, dx$

I have to solve this irrational integral $$ \int \frac{1+\sqrt{x^2+3x}}{2-\sqrt{x^2+3x}}\, dx$$ It seems that the most convenient way to operate is doing the substitution $$ x= \frac{t^2}{3-2t}$$ according to the rule, obtaining the integral: $$…
Anne
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Evaluating an integrals by appropriate substitution

i can't understand how to solve this issue: using an appropriate substitution, evaluate this integral: $$ \int \frac{1+x²}{\sqrt{1+x}}\mathrm{d}x $$ can any one solve this so i can understand how to do this.
Nimi Estela
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Simple explanation of difference between $\cos(x)$ and $\cos(x^2)$ integrals.

I'm not looking for the solved integrals, I'm just looking for a simple explanation for why $\displaystyle\int\cos(x^2)dx$ is so much more complicated than $\displaystyle\int\cos(x)dx$ With simple I mean something I can say to explain the…
Mattis
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How to calculate $ \int \limits_{0}^{\infty} \frac{ t^2 e^{-t}}{(1 + e^{-t})^2} dt $?

I am trying to calculate $$ \int \limits_{0}^{\infty} \frac{ t^2 e^{-t}}{(1 + e^{-t})^2} dt $$ I used variable replacement $u = e^{-t} $ and got the integral $$ \int \limits_{0}^{1} \frac{ \ln^2u}{(1 + u)^2} du $$ Wolfram says that it is…
GThompson
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What property allows me to integrate a gaussian function?

Whenever I integrate a gaussian function, I get to a step that makes me a little uncomfortable because I don't fully understand it. The only way I know of to analytically integrate the gaussian function is to multiply two of them together, like…
Cactus BAMF
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A seemingly simple integral

How would one show the following equality, $$\int_{\mathbb{R}^d}|x|e^{-\frac\beta 2 |x|^2}dx = c_d\beta^{-\frac{d+1}{2}}.$$ Where $c_d$ is a constant only depending on $d$. This is problem is relatively easy to do when $d=1$ by either substitution…
THIG
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Integrating $\int x \arcsin x \,dx$ by parts

Solving the integral $$\int x \arcsin x\,dx$$ I know I have to solve integral using by parts $$\int u\,dv = uv- \int v\,du$$ $$\int x \arcsin x \,dx$$ Let $u = \arcsin x$ and $dv = x\,dx$ so $du = \dfrac1{\sqrt{1-x^2}} \, dx$ and $dv = x^2$. but…
Amy R.
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Integration containing Dirac measure

The Dirac measure is defined by $$\delta_x(A)= \begin{cases} 1 &\text{if $x \in A$}\\ 0 &\text{if $x \notin A$}\\ \end{cases}$$ Now, since $$\int f(y) \, d\delta_x(y)=f(x)$$ where $f:X\rightarrow \mathbb{R}$ is a function. So if $f(x)=1, \forall…
gbd
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Is there a closed form for $\int_0^\infty \frac{\sin(x)}{x^2+a^2} dx$?

As the title says, is there a closed form for $$ \int_0^\infty \frac{\sin(x)}{x^2+a^2} dx \,? $$ The one with $\cos(x)$ instead of $\sin(x)$ can be calculated via a simple application of the Residue theorem, but the computation uses the fact that…
N. S.
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Integral over a ball

Let $a=(1,2)\in\mathbb{R}^{2}$ and $B(a,3)$ denote a ball in $\mathbb{R}^{2}$ centered at $a$ and of radius equal to $3$. Evaluate the following integral: $$\int_{B(a,3)}y^{3}-3x^{2}y \ dx dy$$ Should I use polar coordinates? Or is there any tricky…
Paweł Orliński
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