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.

73636 questions
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Calculate $\int_{\mathbb R^n}\frac{1}{(1+\|x\|_2^n)^2}dx$

How can I calculate the integral $$ \int_{\mathbb R^n}\frac{1}{(1+\|x\|_2^n)^2}dx $$? Is there a "simple" way?
Robert
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Integrate $\frac{x}{\sin 2x}$

\begin{align}\int_{{\pi}\over5}^{{3\pi}\over10}\frac{x}{\sin2x}\,dx\end{align} This integral came up while learning integration using Leibnitz rule. What has been tried is taking the integral…
Shimura Variety
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Solving the integral of $\cos^2x\sin^2 x$

Solving the integral of $\cos^2x\sin^2 x$: My steps are: $(\cos x\sin x)^2=\left(\frac{\sin(2x)}{2}\right)^2$. Now we know that $$\sin^2(\alpha)=\frac{1-\cos (2x)}{2}\iff\left(\frac{\sin(2x)}{2}\right)^2=\frac 14\sin^2(2x)=\frac 14\cdot \frac{1-\cos…
Sebastiano
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Should I integrate constant with x or t while solving integral using substitution?

I am high school student and found a question in NCERT Class 12 book. The question is: $$\int\frac{1}{1-\tan(x)}dx$$ After simplifying the question, I got: $$\frac{1}{2}\left(\int\frac{\cos(x)+\sin(x)}{\cos(x)-\sin(x)}+1\right)dx$$ I then used…
Random Person
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How was the precession of Mercury's orbit calculated in the days before computers?

Over the weekend, I wrote a little program that simulates the solar system (or any set of bodies). I use off-the-shelf orbital elements to set the starting conditions and then do a simulation of gravity to move the bodies, rather than just…
Scott Hurst
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I got different answer of $~\int \frac{1}{(x^2+1)^2}\mathrm{dx}$

$$\begin{align} &\int {1 \over (x^2+1)^2 } \mathrm{dx} ~~ \leftarrow~~ \text{I assume}~~x\ne0 ~~ \text{since it is trivial as it is held} \\ &=\int {x^2+1-x^2 \over (x^2+1)^2 } \mathrm{dx}\\ &=\int \left\{ {(x^2+1) \over (x^2+1)^2 }- {x^2 \over…
electrical apprentice
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Existence of function $f$ with integral properties $\int_{-1}^1 f(x) (1 - |x|) = r, \int_{-1}^1 f^k(x) (1 - |x|) = r^{k + 1}$?

Let $r > 1$ and fix an integer $k>1$. I am wondering if there exists a function $f \colon [-1, 1] \to \mathbb{R}_+$, such that $$ \int_{-1}^1 f(x) \big(1 - |x|\big)\, \mathrm{d} x = r, \quad \mbox{and} \quad \int_{-1}^1 \big(f(x)\big)^k \big(1 -…
Drew Brady
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How to Describe the Period of Motion of a Quartic Potential by Elliptic Integral of First Kind

This is the content of section 1.4 "Instantons and Large N" by Marcos Mariño. The (inverted) potential is following form. $$ V(q)=-\frac{1}{2}q^2+\frac{1}{4}q^4 $$ I want to show the period of…
Anzu Ariake
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Integral of $\frac{2}{x^3-x^2}$

How can I integrate $\dfrac2{x^3-x^2}$? Can someone please give me some hints? Thanks a lot!
darkchampionz
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How did my Professor come up with this substitution?

So my professor gave us this solution to the following integral: $\int \frac{dx}{x\sqrt{x^2-1}}= \int \frac{xdx}{x^2\sqrt{x^2-1}}$ Here he substituted $t=\sqrt{x^2-1}$ from which $t^2=x^2-1$, but what I don't understand is why he then wrote…
downmath
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Evaluate the triple integral $\int _0^1\int _0^1\int _0^1\frac{\ln ^n\left(xyz\right)}{1-xyz}dxdydz$

How can we prove this integral? $$\int _0^1\int _0^1\int _0^1\frac{\ln ^n\left(xyz\right)}{1-xyz}dxdydz=\frac{\left(-1\right)^n}{2}\Gamma \left(n+3\right)\zeta \left(n+3\right)$$ I have received a solution for above Integral from one of my friends…
Asmat Qatea
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Does the following definite integral exist?

I encounter a problem in which I would need to deal with the folloing definite integral $$I(t)=\int_{0}^{\infty} \mathrm dp \frac{p^2}{\omega^5} \sin^2\left(\frac{\omega t}{2}\right)$$ in which $$\omega=\sqrt{m^2+p^2}$$ where m is just some…
Tan Tixuan
  • 161
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Calculate $\iint_S \sin(x+y)dxdy$

Let S be the region bounded by the lines $y=x, x+y = \frac{\pi}{2}, y=0$ and calculate the double integral $$\iint_S \sin(x+y)dxdy$$ I have sketched the region and get the following: I'm unsure as to whether I integrate with respect to the lower…
tesla john
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Finding limits of integration for volume integral

Compute the volume of solid enclosed between the surfaces $x^2+y^2=9$ and $x^2+z^2=9$ What should be the limits of the integrals ? I am getting this z from $-\sqrt{9-x^2}$to $\sqrt{9-x^2}$ x from $-\sqrt{9-y^2}$ to $ \sqrt{9-y^2}$ y from $0$ to…
Aman Mittal
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Separating double integral

Let's say I have the following integral: $$\int_0^1\int_1^3 x+y\: dy\,dx $$ Can I separate this (like with multiplication, i.e x*y) to: $$\int_0^1 x\:dx +\int_1^3 y\: dy $$ Is this correct?
Dakalaom
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