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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How do you do partial fraction $\frac2{x^2 - 8}$?

How do you do partial fraction on $\frac2{x^2 - 8}$ ? Or are there other method of doing? I tried trig substitution but could not get the answer.
sage
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a triple cumulative integration

$$\int_{0}^{2} \mathrm{~d} z \int_{0}^{\left(2 z-z^{2}\right)^{\frac{1}{2}}} \mathrm{~d} y \int_{0}^{\left(2 z-z^{2}-y^{2}\right)^{\frac{1}{2}}}\left(x^{2}+y^{2}+z^{2}\right)^{-\frac{1}{2}} \mathrm{~d} x$$ By $$\int \frac{\mathrm{d}…
jhbihb
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Integration: First fundamental theorem of calculus application

$f(x) = \frac{d}{dx} \int_{x^3}^{x^2} (t^2 +1) dt$, Find the explicit expression of $f(x)$ $\int_{x^3}^{x^2} (t^2 +1) dt = \int_{x^3}^0 (t^2 +1) dt + \int_{0}^{x^2} (t^2 +1) dt = \int_0^{x^2} (t^2+1) dt -\int_0^{x^3} (t^2+1) dt$ However, if I…
user307640
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Integral involving Legendre polynomials and simple rational function

I would like to know if there is a way to express the following integral in terms of known functions $$ I(\ell,a):=\int_{-1}^{1}\frac{P_{\ell}(x)}{x^2+a^2}\mathrm{d}x $$ with $a\in \mathbb{R}$ where $P_{\ell}$ is a Legendre polynomial of order…
user12588
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$\iiint\limits_{D}\frac{dxdydz}{\sqrt{x^2+y^2+(z-\frac{1}{2})^2} }$ D is given by $x^2+y^2+z^2 \leq 1$

$$\iiint\limits_{D}\frac{dxdydz}{\sqrt{x^2+y^2+(z-\frac{1}{2})^2} }$$ D is given by $x^2+y^2+z^2\leq1$ I try to use $ \left\{\begin{matrix} x=r\sin \phi \cos \theta \\ y=r\sin \phi \sin \theta \\ z=r\cos\phi \end{matrix}\right. $ while the…
liyushu
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Evaluating $\int_1^{\pi/2}\frac{\sin(x)\ln(x)}{x^2}dx$

The question goes: Evaluate $\int_1^{\pi/2}\frac{\sin(x)\ln(x)}{x^2}dx$. I have tried by parts, substitutions and other techniques I have learned from first year...but I didn't get any thing look good...
Lalala
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Finding anti-derivative of $f(x)= \sin^3 x \cos^2 x $

Finding anti-derivative of $f(x)= \sin^3 x \cos^2 x $ so, integrate $\int \sin^3 x \cos^2 x dx = \int \sin x (1- \cos^2 x) \cos^2 (x) dx $ let $u = \cos x$ $\int -u^2 (1-u^2) du = \int -u^2 dx + \int u^4 dx = \frac{-u^3}{3} + \frac{u^5}{5} + C…
user307640
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Finding the area of the region bounded by $y=5\csc\theta\cot\theta$, $\theta=3\pi/4$, and $y=5\sqrt{2}$

I am trying to find the area of the shaded area: I formed a rectangle with width of $10\sqrt{2}$ and length of $\frac{\pi}{2}$ Then I need to subtract from the area under graph of $y=5 \csc x \cot x$ To tackle this, I solve part by part…
user307640
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finding integral of $ \int \frac{1}{\sin x + \sqrt{3} \cos x}\ dx $

In $ \int\frac{1}{\sin x + \sqrt{3} \cos x}\ dx $, If I multiply and divide by $1/2$ I get $$ \int\frac{1/2}{\frac{1}{2} \sin x + \frac{\sqrt{3}}{2} \cos x}\ dx ,$$ then I can write $ 1/2=\sin (\frac{\pi}{6}) $ and $ \sqrt{3}/2=\cos (\frac{\pi}{6})…
Manu Sm
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Where to Suggest a New Integral for a Table of Integrals?

I recently derived a closed form for an integral I haven't seen anywhere, $$\int_0^zY_0(\sqrt{z^2-x^2})dx=\frac2\pi\big(\operatorname{Ci}(z)\sin(z) - \operatorname{Si}(z)\cos(z) \big), \qquad z>0,$$ which, numerically, appears to be generalizable to…
astronautgravity
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Improper integral of an even power of $x$ times $e^{-x^2}$

In Donald McQuarrie’s Mathematical Methods for Scientists and Engineers, he has a problem I would like to assign to my class, but I am having trouble solving it. It states Show that $$\int_0^\infty e^{-x^2} \cos\alpha x \,dx = \frac{\sqrt{\pi}}{2}…
Gary L. Gray
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Finding the area under an arc of a circle

The image below shows a circle of radius r with the bottom right quadrant arc in red. Is there an equation for this and only this arc? The reason for the question: what is the area under that arc, highlighted in green? The easy answer…
charlie_sar
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div($F$)=$0$ and $\int_{\partial B_1(0)}\langle F(x),x\rangle dS_{\partial B_1(0)}=1$

Is there a function $F\in C^1(\mathbb R^3,\mathbb R^3)$ with div($F$)=$0$ and $\int_{\partial B_1(0)}\langle F(x),x\rangle dS_{\partial B_1(0)}=1$? This looks a little bit like the theorem of Gauß but I dont know how to prove or disprove it.
andy
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Evaluate $\int^\infty_{-\infty}\frac{1}{x^2-2x\cot(x)+\csc^2(x)}dx$

This question comes from the MIT Integration Bee 2022 (Semi-Final). It can be simplified as $$\int^\infty_{-\infty}\frac{1}{(x-\cot(x))^2+1}dx$$ or $$\int^\infty_{-\infty}\frac{\sin^2x}{x^2\sin^2x-x\sin2x+1}dx$$ The official website gives the answer…
HeyFan
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How to prove $\lfloor\log_{10}\int^\infty_{2022}10^{-x^3}dx\rfloor=-2022^3-8$?

This question comes from the MIT integration Bee 2022 Final Round. As $10^{-x^3} = e^{-x^3\ln10}$, and by substitution $u=x^3\ln10$, the integral becomes $$\int^\infty_{2022^3\ln10}\frac{1}{3(\ln10)^{1/3}}u^{-2/3}e^{-u}du$$ However, I don't know how…
HeyFan
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