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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Show that $\int_{\pi/6}^{\pi/4} \cot x dx= 1/2 \ln 2$

I took $\cot x$ as $\cos x$ divided by $\sin x$ . Substituted $u = \sin x$ , $dx = 1 / \cos x du$. Got $\ln u$ . Replaced $u = \sin x$ and put the limits in . Got $\ln \sqrt{ 2}$. What should I do next? .
Lilly
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Solving the following integral (rational function, cubic over linear)

I'm trying to solve the following integral: $$ \int\frac{x^3}{(x+2)}\mathrm{d}x $$ It would seem to me to be a classic integration-by-parts problem, but trying to do that (with $u=x^3$ and $dv=1/(x+2)$ I find myself stuck with the integral: $$ \int…
user159527
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Using Stokes' Theorem to evaluate an integral around a triangular path

Problem: Use Stokes’ theorem to evaluate the integral $I = \int\limits_C \textbf{F} \centerdot \textbf{ds}$ when $\textbf{F}$ is the vector field $\textbf{F} = 3zx\textbf{i} + 3xy\textbf{j} + yz\textbf{k}$ and C is the path consisting of the three…
Swamp G
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How to compute $\int \frac{x}{(x^2-4x+8)^2} \mathrm dx$?

Can someone help me to compute: $$\int \frac{x}{(x^2-4x+8)^2}\mathrm dx$$ And, in general, the type: $$\int \frac{N(x)}{(x^2+px+q)^n}\mathrm dx$$ with the order of polynomial $N(x)
bleish
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Relationship between Archimedean spiral and Theorem of Pythagoras

I am interested to find a particular connection between a right triangle and an arithmetic spiral. In a right triangle, when one side is $0.8$ and the other side is $0.6$, the hypotenuse is $1$. Trivially, when one side is $1$ and the other side is…
argonaut
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Integral of $x\sin^2 (x^2)$

I'm trying to find the integral of $y = x\sin^2 (x^2)$. Can someone help please? I've tried converting it to $x(\frac{1}{2} -\frac{1}{2}\cos(2x^2))$ and using integration by parts but it doesn't seem to help. Thanks.
thbcm
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Area for an ellipsoid

"Calculate the area for the rotation ellipsoid you get by rotating the ellipsoid $\frac{x^2}{2}+y^2 = 1$ around the x-axis." I solved for x: $$ y = \pm \sqrt{1-\frac{x^2}{2}} $$ Then did $ y = 0$ to get the integration limits, $\pm \sqrt(2)$. So…
iveqy
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What is this equation with zeta from a T-Shirt in a video?

There's an equation on a T-shirt in the music video by Remy Zero for "Gramarye". There's not a completely clear shot of it, but it's something along the lines of: $$?^{???}(z) = \frac{n !}{2 \pi ?} \int_{C} \frac{f(\zeta)}{(\zeta - z)^{n+1}} d\zeta…
HostileFork says dont trust SE
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Evaluate $\int x e^{\sqrt{x}} \, dx$

$$\int_0^1 xe^{\sqrt{x}} dx = ? $$ All I can think of is the integration by parts rule, where $ u = x $ and $ dv= e^{\sqrt(x)} $ $ \Rightarrow du = 1$ and $ v= e^{\sqrt(x)} $ The answer I get is $e^{\sqrt(x)}(x-1)$ , which is wrong. Can anyone…
Minu
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Multidimensional integral involving delta functions

The question is to compute the following multidimensional integral: \begin{equation} \omega^{(T)}({\bf c}) := \int\limits_{{\mathbb R}^{2 T}} \delta\left( c_{1,1} - \sum\limits_{j=1}^T x_{1,j}^2 \right) \delta\left( c_{2,2} - \sum\limits_{j=1}^T…
Przemo
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Let $E=\{1/n:n\in\mathbb{N}\}$. Prove that a function is integrable on $[0,1]$.

Would someone be kind enough to explain to me what the notation in this exercise means? This is only one exercise of 1/12 of a course, but I really want to know what it is. Thanks! So the problem says: "Let $E=\{1/n:n\in\mathbb{N}\}$. Prove that the…
s1047857
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$\int \frac{dx}{x+\sqrt{x}}$

Please, help me understand how to find $$\int \frac{dx}{x+\sqrt{x}} = 2 \ln(\sqrt{x} + 1)$$ Is it done by some kind of substitution? Note: by integrating the LHS, not differentiating RHS.
kooce
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How to prove $ \int _0 ^1 \left ( \sqrt{1-x^2}\right )^n dx = \prod_{k=1} ^n \frac {2k}{2k+1} $

How to prove this integral identity? $$ \int _0 ^1 \left ( \sqrt{1-x^2}\right )^n dx = \prod_{k=1} ^n \frac {2k}{2k+1} $$ ↑ This identity is false. It should be corrected to $ \int _0 ^1 \left ( {1-x^2}\right )^n dx = \prod_{k=1} ^n \frac…
SJR
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How find this integral $I(a,b)=\iiint_{x^2+y^2+z^2\le 1}(ax+by)^2\,dx\,dy\,dz$

Find this integral $$I(a,b)=\iiint_{x^2+y^2+z^2\le 1}(ax+by)^2\,dx\,dy\,dz$$ since $$(ax+by)^2=a^2x^2+b^2y^2+2abxy$$ so $$I=I_{1}+I_{2}+I_{3}$$ where $$I_{1}=\iiint_{x^2+y^2+z^2\le 1}ax^2\,dV=a\iiint_{x^2+y^2+z^2\le…
math110
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Understanding an issue in a triple integral solution

Here I have a triple integral $$ \iiint f(x,y,z)dxdydz $$ on the region : $\{\sqrt[]{x^2+y^2} \le z \le \sqrt[]{4-x^2-y^2}\} $ if we use cylindrical coordinates we have : (1) $ r\le z \le \sqrt[]{4-r^2} $ and when we want to do the integral…
Waseem Francis
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