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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An equivalent of : $f(x)=\int_0^{+\infty}\frac{e^{-xt}}{(1+t^3)^{1/3}} dt$

$\forall\ x\ \in\ \left]0,+\infty\right[\ $ we put: $$ {\rm f}\left(x\right) = \int_{0}^{\infty}{{\rm e}^{-xt} \over \left(1 + t^{3}\right)^{1/3}}\,{\rm d}t $$ The question is the question is to find an equivalent of $\,\,{\rm f}\left(x\right)$ when…
Mohamed
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Limitations and alternatives to Riemann Integral

Why are there integrals different from Riemann's? That is, what aspect of a function makes it non-Riemann integrable but integrable by other approaches, what are the other approaches, and are their results differentiable or do they not have inverse…
George Frank
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How to prove $\frac{1}{\pi}\int_{0}^{\pi}\frac{\sin{t}}{\cos{t}-\cos{x}}f'(x)\,dx=\frac{1}{2\pi}\int_{0}^{2\pi}\cot{\frac{x-t}{2}}g'(x)\,dx$?

show that $$\dfrac{1}{\pi}\int_{0}^{\pi}\dfrac{\sin{t}}{\cos{t}-\cos{x}}f'(x)dx=\dfrac{1}{2\pi}\int_{0}^{2\pi}\cot{\dfrac{x-t}{2}}g'(x)dx$$ where $g$ denotes the odd and $2\pi$ periodic extension of $f$ onto all of $R$. My try:…
math110
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Showing integral is bounded

Let $A \subset \mathbb{R}^n$ be open and bounded. Is it true that $$\int_A \int_A |x-y| \, dx \, dy \leq R|A|^2$$ where $R$ is some number (eg. the radius of a ball containing $A$) and $|A|$ is measure of $A$? How do I show this rigourously?
Student101
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Trigonometric Integral : $\int\frac{1}{\sin x+ 3\cos x}dx$

I would appreciate if somebody could help me with the following problem Q: How to integrate this integral $$\int\frac{1}{\sin x+ 3\cos x}dx$$
Young
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Calculus integration of the Gaussian distrib. bell curve??

$$y=\frac{1}{\sqrt{2\pi}}e^{-x^2/2}$$ Looking at the Gaussian distrib. function (bell curve) Is this an impossible integration? http://www.wolframalpha.com/input/?i=%5Cfrac%7B1%7D%7B%5Csqrt%7B2%5Cpi%7D%7De%5E%7B-x%5E2%2F2%7D
JackOfAll
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Interpretation of an integral involving two curves

I'm currently enrolled in a course dealing with Isogeometric Analaysis (Splines, NURBS, ...) and in one of our exercises, we are asked to numerically evaluate an integral using methods we have implemented. This works fine for me, but I actually…
Thomas
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Integration of parabola

I have this homework question I am working on: The base of a sand pile covers the region in the xy-plane that is bounded by the parabola $x^2 +y = 6$ and the line $y = x$: The height of the sand above the point $(x;y)$ is $x^2$: Express the volume…
Raynos
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Integral of $\int \frac{\sin^22x}{\cos2x}\,\mathrm dx.$

I would like to solve the following integral $$\int \frac{\sin^22x}{\cos2x}\,\mathrm dx.$$ I know that the derivative of $\cos$ its $\sin$ but how its help me? Any suggestions? Thanks.
Ofir Attia
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Indefinite Integral of $(x^2+1)\over (x^3 + 8)$

I need help in solving this indefinite integral: $$\int {(x^2+1)\over (x^3 + 8)} $$ I know it needs to be reduced to the form $A\over (x+2)$ + $B\over (x+2)^2$ where A and B are constants, but I cannot seem to solve this integral. Thanks in…
user110441
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Evaluating $\sum_{n=1}^{\infty} \int_{0}^{\pi}{\cos x \cos nx \over \cos^2x+h_1^2}dx\int_{0}^{\pi}{\sin x \sin nx \over \cos^2x+h_2^2}dx$.

How to evaluate the integral $$\displaystyle\sum_{n=1}^{\infty} \int_{0}^{\pi}{\cos x \cos nx \over \cos^2x+h^2}dx \int_{0}^{\pi}{\sin x \sin nx \over \cos^2x+h^2}dx$$ and $$\displaystyle\sum_{n=1}^{\infty} \int_{0}^{\pi}{\cos x \cos nx \over…
sam
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Price-Demand, Marginal-price and other financial jargon

So my book likes to assume that I already have a business degree while learning calculus so I need your help to clarify my book's questions. It asks: Price-demand equation. The marginal price for a weekly demand of x bottles of shampoo in a…
Paze
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How find this $\int_{1}^{\infty}\frac{1}{x^2\sqrt{x^3-1}}dx$

find The integral $$\int_{1}^{\infty}\dfrac{1}{x^2\sqrt{x^3-1}}dx$$ My try:let $$\sqrt{x^3-1}=t\Longrightarrow x^3=t^2-1$$ Thank I can't,I think this answer maybe use Gamma integral
user94270
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Tricky Integral Involve Sine

How would I integrate the following: $$ \int \frac{c}{\sin(t)\sqrt{\sin^2(t) - c^2}} \, dt $$ Here $c$ is a constant. I have tried numerous substitutions, but I just can't seem to get the right one. Integration by parts does not seem to be of any…
John Sweeney
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Integral Of $\int \frac{2\cdot \cos^2(x)}{x^2}dx$

I`m trying to integrate the following: $$\int\frac{2\cdot \cos^2(x)}{x^2}dx$$ what I did first is: $$\int \frac{2\cdot (\frac{1}{2}+\frac{cos2x}{2})}{x^2}dx=\int \frac{1+cos2x}{x^2}dx$$ now what? any suggestions? thanks!
Ofir Attia
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