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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Convergence of $\int\int_{|x|\geq 1,|y|\geq 1} \frac{1}{|x|^\alpha+ |y|^\beta} \;dx\;dy$

For which values of $\alpha$ and $\beta$ does the following integral converge? \begin{equation} \int\int_{|x|\geq 1, |y|\geq 1} \frac{1}{|x|^\alpha+ |y|^\beta} \; dx \; dy, \quad \alpha,\beta>0. \end{equation} Thank you!
Berkheimer
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How to evaluate $\displaystyle\int {1\over (1+kx^2)^{3/2}}dx$

What change of variable should I use to integrate $$\displaystyle\int {1\over (1+kx^2)^{3/2}}dx$$ I know the answer is $$\displaystyle x\over \sqrt{kx^2+1}.$$ Maybe a trig or hyperbolic function?
kiddo
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How to determine if $\int_2^\infty x^2/e^x dx$ converges without computing it?

How to determine if $\int_2^\infty x^2/e^x \; dx$ converges without computing it? I'm thinking of applying a comparison test but I'm not sure to what.
user93200
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How to evaluate $\int \sin(x)\arcsin(x)dx$

I need to evaluate following integral $$\int \sin(x)\arcsin(x) \ dx$$ Can anyone please help me? Thanks.
Aven
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Evaluating the definite integral $\int_0^\infty \frac{\mathrm{e}^x}{\left(\mathrm{e}^x-1\right)^2}\,x^n \,\mathrm{d}x$

I am having difficulty evaluating definite integrals of the form $\int_0^\infty \frac{\mathrm{e}^x}{\left(\mathrm{e}^x-1\right)^2}\,x^n \,\mathrm{d}x$. I would appreciate any guidance that could be offered. I am aware that these evaluate to constant…
user001
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Change of variables

How can I convert the integral $\int_0^{2\pi} (a^2 \cos^2 t +b^2\sin^2 t)^{-1} dt$ into an integral $\oint_\gamma z^{-1} dz$ where $z\in \mathbb C$ and $\gamma: {x^2\over a^2}+{y^2\over b^2}=1$? I can see that $|z|^2 = a^2 \cos^2 t +b^2\sin^2 t$ but…
Gore
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How can I write fun, cool, and challenging integration problems?

I just asked this question on another site but with only a handful of somewhat helpful responses. I just made the following integral problem which I think is very cool and nice, but I don't know how to solve it. (I found it by taking the derivative…
Faraz Masroor
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If I have an integral equality and then I multiply a variable inside the integral, does the equality still hold?

So say for example I have $\int_{x=a}^{b}A(x)dx = \int_{x=a}^{b}B(x)dx$ for some functions $A(x)$ and $B(x)$. Then does $\int_{x=a}^{b}xA(x)dx = \int_{x=a}^{b}xB(x)dx$ still hold ? I was solving another problem and I used this without really…
Donova
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Integrate $\int\sqrt{1+9t^4}\:dt$

I need to find \begin{align} \int\sqrt{1+9t^4}\:dt. \end{align} What I have so far: \begin{align} \int\sqrt{1+9t^4}\:dt & =\int\sqrt{1+\left(3t^2\right)^2}\:dt,\tag{1} \end{align} now let $3t^2=\tan\left(\theta\right)\implies \displaystyle…
bjd2385
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Indefinite integral of $\arctan{\sqrt{1-x^{2}}}$

All is in the title: what is the antiderivative of $x\mapsto \arctan{\sqrt{1-x^{2}}}$ ? I'm supposed to tutor younger students taking an integration class, and this is one of their exercises. I strongly dislike this kind of math and, consequently,…
Sergio
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Integration help - question: $e^{-\sin(x)}$

I would really like some help with the integration of $e^{-\sin(x)}$. Thanks to anyone who will help :) Given that $\sin(x) > \frac{2x}{\pi}$ for $0 < x < \frac{\pi}{2}$, where $$\int_0^{\pi/2}e^{-\sin x}\,dx<\int_0^{\pi/2}e^{-2x/\pi}\,dx$$ RTS:…
betty_carr
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A tricky double integral

What is $$\int_0^1 \int_0^1 \frac{ dx \; dy}{1+xy+x^2y^2} ? $$ Can you do one of the integrals and turn it into a single integral? I get lost in a sea of inverse tangents.
Herman
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Integration with complex numbers

I know that you can integrate $$\int e^{-x}\cos(x)dx$$ by parts, but I would like to know how you can use complex variables instead.
Julieta Molina
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Integral of $(\log x)^3$ (Spivak's Calculus, Chapter 19, Problem 3v)

$$\int (\log x)^3\,dx$$ I solved the problem by applying integration by parts twice: First, I write the integrand as $(\log x)^2\cdot\log x$, and then I set $u=(\log x)^2 \Rightarrow du=\frac{2\log x}{x}dx$ and $dv=\log x\,dx \Rightarrow v=x\log…
rae306
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How to solve the integral: $\int {\sqrt{1+\sqrt{x}}}/x dx$

$$\int \frac{{\sqrt{1+\sqrt{x}}}}{x} dx$$ I tried with $u=\sqrt x $, but this did not work. I really don't know what to do...
ParaH2
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