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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Explanation for $\int_{\frac{\pi}{2}+(j-1)\pi}^{\frac{\pi}{2}+j\pi} \frac{|\cos(t)|}{\frac{\pi}{2}+j\pi} \,dt = \frac{2}{\frac{\pi}{2}+j\pi} $

I don't see how / why one can rewrite the integral as following: $$\int_{\frac{\pi}{2}+(j-1)\pi}^{\frac{\pi}{2}+j\pi} \frac{|\cos(t)|}{\frac{\pi}{2}+j\pi} \,dt = \frac{2}{\frac{\pi}{2}+j\pi} $$ I think this should be rather easy, but I don't see…
user866761
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Evaluating $\int_0^\infty \frac{e^{-tx^2}-e^{-x^2}}{x}dx$

I have been working on the following integral: $$\int_0^\infty \frac{e^{-tx^2}-e^{-x^2}}{x}dx$$ where $t$ is any nonnegative real number. Would anyone be able to provide a hint or provide a solution on how such an integral should be approached? I…
CBBAM
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Integration of $ \cos x.\cos 2x...\cos nx$

I wanted to integrate $\int \cos x\cos 2x\cdots \cos nx \, dx$. What I know is that $ \cos x\cos 2x\cdots \cos nx=\dfrac{1}{2^{n-1}}\sum_\pm \cos((n\pm(n-1)\pm\cdots\pm2\pm1)x)$ where the sum is over all $2^{n-1}$ possible $\pm$. But quite obviously…
Manjoy Das
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Comparing the value of $\int _0 ^{\frac{\pi}{2}} x^n \sin^nx\,dx$ for different $n$

Let $$I_n = \int _0 ^{\frac{\pi}{2}} x^n \sin^nx \, dx.$$ Could you tell me how to decide which one is true: $I_{2012}>I_{2013} $, or $I_{2013}>I_{2012}$?
Hagrid
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Finding the nth integral of $\ln(1+z/k)$

I can find the nth integral of $\ln(z)$ as follows: \begin{aligned} \left(\frac…
Spador Yedi
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Evaluate $\int_0^{\pi/2} \frac{1+2\cos x}{(2+\cos x)^2}$

I don’t know how to begin solving. Can I get a hint? My failed attempt Let $t =\tan (x/2)$ Then $$I =2 \int_0^1 \frac{(3-t^2)}{(3+t^2)^2 }$$ which I am not able to solve
Aditya
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Proof for an integral identity

Is it true that $\int_0^A dx \int_0^B dy f(x) f(y) = 2 \int_0^A dx \int_0^x dy f(x) f(y)$ ? If so, can this be proved?
Raghunathan Ramakrishnan
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How to prove $\int_{0}^{\infty}\frac{\tan^{a} (x) \;dx}{x^2+b^2}=\frac{\pi}{2b}\frac{\tanh^{a} (b)}{\cos\frac{a\pi}{2}}$

I'm having trouble to prove the following: $$ \int_{0}^{\infty}\frac{\tan^{a}\left(x\right)} {x^{2} + b^{2}}\,\mathrm{d}x = \frac{\pi}{2b}\,\frac{\tanh^{a}\left(b\right)}{\cos\left(\pi a/2\right)}\quad \mbox{where}\ \left\vert a\right\vert <…
Martin Gales
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Integration involving roots

$$\int\frac{dx}{(1+x^\frac{1}{4})x^\frac{1}{2}}$$ This is my work: $$u^4=x$$ $$4u^3=dx$$ $$\int\frac{4u^3du}{(1+u)u^2}=\int\frac{4u^3du}{(1+u)u^2}=-4(1+x^\frac{1}{4})^{-1}+2(1+x^\frac{1}{4})^{-2}+C$$ But wolframAlpha gives quite a different answer.…
user1242967
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Integral of $ \int_{-1}^{1} \frac{x^4}{x^2+1}\,dx $

Any suggestions how to solve it? by parts? $$ \int_{-1}^{1} \frac{x^4}{x^2+1}dx$$ Thanks!
Ofir Attia
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Using area=$\int_{y = a}^{y = b} x \, dy $ find the following shaded area:

So I did: $$\eqalign{ & y = {1 \over x^2} \cr & {x^2} = {1 \over y} \cr & x = y^{ - {1 \over 2}} \cr & \int_a^b x \, dy = \left[ 2{y^{1 \over 2}} \right]_{1 \over 4}^1 \cr & = 2\sqrt 1 - 2\sqrt {1 \over 4} \cr & = 2 - 1 \cr…
seeker
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Integration by parts involving divergence

In Griffiths' E&M, there is an equation that describes energy of a charge distribution as- $W = \frac{\epsilon_0}{2}\int(\nabla.\textbf{E})V d\tau$ The author then performs integration by parts to get- $W = \frac{\epsilon_0}{2}[-\int…
Paddy
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Evaluate $\int {\sin 2\theta \over 1 + \cos \theta} \, d\theta $, using the substitution $u = 1 + \cos \theta $

Evaluate $$\int_0^{\pi \over 2} {{\sin 2\theta } \over {1 + \cos \theta }} \, d\theta$$ using the substitution $u = 1 + \cos \theta $ Using $$\begin{align} u &= 1 + \cos \theta \\ \frac{du}{d\theta} &= -\sin\theta \\ d\theta &=…
seeker
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How to solve this integral $I = \int\dfrac{\cos^3x}{\sin x + \cos x}dx$?

$\displaystyle\int\dfrac{\cos^3x}{\sin x + \cos x}dx$ I added $J =\displaystyle \int\dfrac{\sin^3x}{\sin x + \cos x}dx$ then $I + J = \displaystyle\int\dfrac{\cos^3x + \sin^3x}{\sin x + \cos x}dx = x + \dfrac{1}{2}\cos2x + C$ but I can't find how to…
Solitarie
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Integral Of $\int\sqrt{\frac{x}{x+1}}dx$

I want to solve this integral $$\int\sqrt{\frac{x}{x+1}}dx$$ And think about: 1) $t=\frac{x}{x+1}$ 2) $dt = (\frac{1}{x+1} - \frac{x}{(x+1)^2})dx$ Now I need your advice! Thanks!
Ofir Attia
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