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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Integral inequality: $\frac{1}{30}<\int_{2}^{\infty}\frac{\sqrt{s^3-s^2+3}}{s^5+s^2+1}\,ds<\frac{\sqrt{2}}{20}$

Prove that $$\frac{1}{30}<\int_{2}^{\infty}\frac{\sqrt{s^3-s^2+3}}{s^5+s^2+1}\,ds<\frac{\sqrt{2}}{20}$$ I think this inequality can $s^5+s^2+1>( ) $ and $\sqrt{s^3-s^2+3}<()$?
math110
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Volume between sphere and cylinder

Find the volume of the portion of the sphere $x^2 + y^2 + z^2$=$a^2$ lying inside the cylinder $x^2 +y^2$=ay I think we are supposed to do it in spherical coordinate system but i don't know how to set up limits ...the only thing i can get from the…
Abhinav
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Solving $\int dy f(x,y) g(y,z) = h(x,z)$ for $f$?

Let's say I have an equation of form: $\int_{-\infty}^\infty dy f(x,y) g(y,z) = h(x,z)$ where I've been given $g$ and $h$, and need to find $f$. I shouldn't in general expect a suitable $f$ to exist, or for it to be unique if it does. But does there…
Henry Shackleton
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Calculate:$\int \frac{1}{(x+1)^\frac{3}{4}(x+2)^{\frac{5}{4}}}\ dx$

Calculate following integration $$\int \frac{1}{(x+1)^\frac{3}{4}(x+2)^{\frac{5}{4}}}\ dx$$
kalpeshmpopat
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Can this function be integrated with respect to x?

I would like to integrate the following function with respect to $x$. This is for a signal processing project. $y = \cos(x + ay -a), \ -1 < a < 1$ so the function would pass the vertical line test. Is it possible to integrate this type of…
Auggie
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$f(x) = \int_{k=0}^x e^{f(k)}\, dk$

I have the function $$f(x) = \int\limits_{k=0}^x e^{f(k)} dk$$ How can I solve for an explicit formula of $f$? In my current solution, it becomes undefined past a certain value of $x$ Atempts: Taking a derivative, we have $$f'(x) =…
programjames
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Evaluating $\int \sqrt\frac{x^3-3}{x^{11}} dx$

$$\int \sqrt\frac{x^3-3}{x^{11}}\,dx$$ The form the answer takes suggests a very quick substitution should be possible. I cannot see how to obtain it in only a few steps and would be grateful for any help.
user548941
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Solving $\int\left(\sqrt\frac{gx}{a\ln(x)+d+cx}\right)dx$ using Risch algorithm

I was solving some problems in particle physics, and i got stuck with such an equation, $$\int\left(\sqrt\frac{gx}{a\ln(x)+b+cx}\right)dx$$ The problem was to formulate a function of time and displacement of a particle colliding with it's…
A.M.M Elsayed 马克
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Existence of Elementary Antiderivatives

A version of Liouville's criterion states that: Choose functions $~f~$ and $~g~$ such that $~f≠0~$ and $~g~$ is non-constant. Then, the function $~f(x)e^{g(x)}~$ can be integrated in elementary terms iff there exits a rational function $~R(x)~$ such…
Hello
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Where did I go wrong evaluating this integral?

I wanted to see if complex analysis could help me evaluate the following integral: $$I=\int_{0}^{\infty} \cos(x) e^{−x} dx $$ It definitely converges due to the exponentially decaying function. I began by substituting $\cos(x)$ with $\frac{e^{ix}+…
Joshua Pasa
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Reduction formula benefits

What are the major benefits of the method of the reduction formula?? Aside from the fact that it is fast. Does it really have any advantages over integration by parts and other forms of integration in terms of accuracy?
Orestes Dante
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Searching for an elegant, high school level, technique for solving an integral by hand

I would appreciate some assistance finding alternative approaches to solving the integral posed in the link below. https://www.desmos.com/calculator/rstvj2v3cs It describes and graphs a progression for decomposing the…
Zachary Blakley
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Integral of periodic function on R with sinus cardinal squared

I am looking for a proof that for a function $F$ that is 1-periodic: $$\int_{-\infty}^{\infty}F(x)\operatorname{sinc}^2(\pi x)dx=\int_{0}^{1}F(x)dx$$ edit : with the sinus cardinal function : $\operatorname{sinc} : x \to \frac{\sin(x)}{x}$ if $x…
mocquin
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Integration and domain

Let have for example $$f(x) = \frac{\sin^2 x}{1- \cos x}$$ And let say that we want calculate: $$ \int_{-\pi/2}^{\pi/2} f(x) \, dx $$ After transform: $$f(x) = \frac{\sin^2 x}{1- \cos x} = 1 + \cos x$$ so $$ \int_{-\pi/2}^{\pi/2} f(x) \, dx = x…
user617243
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How to evaluate the integral of $\int\sin(\sin(x))dx$?

I recently watched a video on YouTube talking about something called as horseshoe integrals. A bit of research told me that that's not really a thing. Nonetheless, the YouTuber referred to the integral $$ \int\sin(\sin(x))dx $$ to be an example of…
Archisman Dinda
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