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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This integral evaluates to infinity, does this mean it exists or doesn't exist?

Say I have$f(x) = x^2$ for all $x \in \mathbb{R}$ Does the integral of f(x) over the entire real line exist? It's infinity so does that mean it doesn't exist?
dukenukem
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Can I integrate both sides of this equation?

I have functions continuous functions $f$, $g$ and $h$ on a bounded closed interval. I have an equation $|f - h + h - g| \leq |f - h| + |h - g|$. Can I integrate both sides of this equation over the interval?
Jim_CS
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$\int\frac{x^3}{\sqrt{4+x^2}}$

I was trying to calculate $$\int\frac{x^3}{\sqrt{4+x^2}}$$ Doing $x = 2\tan(\theta)$, $dx = 2\sec^2(\theta)~d\theta$, $-\pi/2 < 0 < \pi/2$ I have: $$\int\frac{\left(2\tan(\theta)\right)^3\cdot2\cdot\sec^2(\theta)~d\theta}{2\sec(\theta)}$$ which is…
cybertextron
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Find the moment of inertia about $x$-axis of the region bounded by $y=x^2$ and $y=x$

Find the moment of inertia about $x$-axis of the region bounded by $y=x^2$ and $y=x$, if the density is proportional to the distance from the $x$-axis. And the answer should represented by the mass of the region. I tried first let the density…
Matata
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how to understand the area of the product of two functions

I just learn something about finding the area enclosed by a function with integral. If there is a function $f(x)$, the integral within $[a,b]$ will the the area that the area enclosed by $f(x)$ in $a$ and $b$ and the x axis. Now, if you have two…
user1285419
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Green's formula (integration in parts)

Let $v,w\in C^1(\bar{\Omega})$ and $\partial\Omega$ be smooth. Then Green's formula in $\mathbb{R}^2$, which is some integration by parts analogon to $\mathbb{R}^1$, is given to be $$ \int_{\Omega}v_{x_i}w\, dx=-\int_{\Omega}vw_{x_i}\,…
Rhjg
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Evaluating $\iiint x\,dx\,dy\,dz$ limited by paraboloid of equation $x=4 y^2+4z^2$ & for plane $x=4$

Hi we have the following problem: $\iiint x\,dx\,dy\,dz$ limited by the paraboloid of equation $x=4 y^2+4z^2$ and for the plane $x=4$. We are having difficulty on finding the limits of each integral and how to turn to polar coordinates. Could…
Edoardo
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How to integrate $\int \frac1{(3+4\sin x)^2}\,dx$?

We are to solve $$\int \frac1{(3+4\sin x)^2}\,dx.$$ I had tried expanding the denominator and substituting $\sin x$ in terms of $\tan(x/2)$ and then putting $\tan(x/2) =t$. But, this made the integration even more complex. I got a large…
Amritansh Singhal
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Integral Arising in 2-Dimensional QED

I have an integral which arises while quantizing QED near a graphene sheet. It is $$\int_{0}^{\infty}\mathrm{d}u\int_{0}^{2\pi}\mathrm{d}\theta\,u\frac{e^{iqu\cos{\theta}}}{\sqrt{u^2+z^2}}$$ Mathematica tells me the answer is $2\pi\,e^{-q|z|}/q$,…
Bob Knighton
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Why is the formula $\ Arc length = \int r \times d\theta $ not correct?

The formula for calculating the area of a curve in a polar graph is $\large \rm \frac{1}{2}\int r^2~ d\theta $ and is adapted from $$\large \rm Area = \frac{1}{2}r^2\times \theta $$ But the formula to calculate the arc length is very different from…
mathnoob123
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Interesting Logarithmic Integral: $\int_{0}^{1} \frac{\ln^2 x \ln^2(1+x)}{x} \;dx $

Other than numerical approximation, how can we calculate the closed form of this integral? $$\int_{0}^{1} \frac{\ln^2 x \ln^2(1+x)}{x} \;dx $$
Aditya Narayan Sharma
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Evaluation of $\int_{0}^{\sqrt{2}-1}\frac{\ln(1+x^2)}{1+x}dx$

Evaluation of $$\int_{0}^{\sqrt{2}-1}\frac{\ln(1+x^2)}{1+x}dx$$ $\bf{My\; Try:::}$ Let $$I(a) = \int_{0}^{\sqrt{2}-1}\frac{\ln(1+ax^2)}{1+x}dx$$ Now $$I'(a) = \int_{0}^{\sqrt{2}-1}\frac{x^2}{(1+ax^2)(1+x)}dx =…
juantheron
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Help with an integral with an infinite sum?

I have the following, very messy integral: $$\int_{0}^{1}\sum_{k=1}^{\infty} \frac{(-\frac{e^x}{x+1}+1)^k}{k} dx $$ Neither WolframAlpha nor Mathematica was of any help. I'm not even sure where to start on this integral - I don't believe that it…
user348212
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How to integrate $\frac {\cos (7x)-\cos (8x)}{1+2\cos (5x)} $ ?

How to integrate $\frac {\cos (7x)-\cos (8x)}{1+2\cos (5x)} $ ? All I could do is apply difference of cosines formula in numerator.After that I'm stuck.Can somebody please help me?
user220382
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$\int\frac{\sin x}{\sqrt{1-\sin x}}dx=?$ Calculate this integral

$\displaystyle\int\dfrac{\sin x}{\sqrt{1-\sin x}}dx=?$ Effort; $1-\sin x=t^2\Rightarrow \sin x=1-t^2\Rightarrow \cos x=\sqrt{2t^2-t^4}$ $1-\sin x=t^2\Rightarrow-\cos x dx=2tdt\Rightarrow…
Micheal Brain Hurts
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