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
20
votes
3 answers

Why are some non elementary integrals defined and others are not?

Why are some integrals that cannot be integrated in elementary terms defined and given names, while others aren’t? Based on what criteria are they chosen? Applicability to real life? And what is the point if we cannot solve them? For…
user372003
  • 521
20
votes
2 answers

how to calculate integral of square of a function

When doing differentiation, I know that if $f$ is a function on $x$, then $$ { d \over dx } f^2 = 2 f {df \over dx} $$ so the opposite in integration is also clear: $$ \int 2 f { df \over dx } dx = f^2 $$ I also know that $$ \int x^2 dx = { x^3…
jamadagni
  • 573
20
votes
2 answers

Integrate Form $du / (a^2 + u^2)^{3/2}$

How does one integrate $$\int \dfrac{du}{(a^2 + u^2)^{3/2}}\ ?$$ The table of integrals here: http://teachers.sduhsd.k12.ca.us/abrown/classes/CalculusC/IntegralTablesStewart.pdf Gives it as: $$\frac{u}{a^2 ( a^2 + u^2)^{1/2}}\ .$$ I'm getting back…
Koobz
  • 303
19
votes
1 answer

Proving a proposition which leads the irrationality of $\frac{\zeta(5)}{\zeta(2)\zeta(3)}$

Question : Is the following $(\star)$ true for $a,b,c\in\mathbb Z$ ? $$\begin{align}\int_{0}^{\frac{\pi}{2}}(ax^4+b\pi x^3+c{\pi}^{2}x^2)\log(\sin x)dx=0\Rightarrow a=b=c=0\qquad(\star)\end{align}$$ Motivation : I've been asking the following…
mathlove
  • 139,939
19
votes
4 answers

How to evaluate this integral? (relating to binomial)

I saw some result that some article used, (without proving) that stated:$$\int_0^1 p^k (1-p)^{n-k} \mathrm{d}p = \frac{k!(n-k)!}{(n+1)!}$$ But I was wondering, how would you integrate it? How did this integral come about? Is it something to do with…
Heijden
  • 331
  • 1
  • 3
  • 12
18
votes
3 answers

Evaluating $\int_{0}^{\infty}\sin^3(x)\cos[a\tan(x)]\frac{dx}{x}$

$$I(a)=\int_{0}^{\infty}\sin^3(x)\cos[a\tan(x)]\frac{dx}{x}$$ I'd like to evaluate the integral by differentiating with respect to parameter $a$ but no success yet. Seems impossible. What would the other options? Edit: A hypothetical closed form…
Martin Gales
  • 6,878
18
votes
1 answer

Is there a non-elementary function with an elementary derivative and an elementary inverse?

Elementary functions are combinations of powers, exponentials and logarithms, using composition and arithmetic operations. The inverse of an elementary function may not be elementary, and the integral of an elementary function may not be elementary.…
Paul Castle
  • 761
18
votes
1 answer

Indefinite Integral of $\sqrt{\sin x}$

$$\int \sqrt{\sin x} ~dx.$$ Does there exist a simple antiderivative of $\sqrt{\sin x}$? How do I integrate it?
Swapnanil Saha
  • 739
  • 1
  • 7
  • 18
18
votes
3 answers

Is this definite integral really independent of a parameter? How can it be shown?

I want to find a nice simple expression for the definite integral $$\int_0^\infty \frac{x^2\,dx}{(x^2-a^2)^2 + x^2}$$ Now, I can numerically compute this integral, and it seems to converge to $\pi/2$ for all real values of $a$. Is this integral…
Keenan Pepper
  • 677
17
votes
2 answers

Why some functions are not integrable?

I know that certain functions are not integrable. Geometrically integration is finding the area under the curve of the graph of the given function. So, in another way why is it not possible to find the area under the curve in case of certain…
Rajath Radhakrishnan
  • 1,715
17
votes
2 answers

$\int_0^{\infty } \frac{\log (x)}{e^x+1} \, dx = -\frac{1}{2} \log ^2(2)$ How to show?

$$ \int_0^{\infty } \frac{\log (x)}{e^x+1} \, dx = -\frac{1}{2} \log ^2(2) $$ Anyone an idea on how to prove this?
wnvl
  • 3,010
16
votes
1 answer

How find this integral $\int_{0}^{\infty}\frac{dx}{(1+x^2)(1+r^2x^2)(1+r^4x^2)(1+r^6x^2)\cdots}$

prove that this integral $$\int_{0}^{\infty}\dfrac{dx}{(1+x^2)(1+r^2x^2)(1+r^4x^2)(1+r^6x^2)\cdots}= \dfrac{\pi}{2(1+r+r^3+r^6+r^{10}+\cdots}$$ for this integral,I can't find it.and I don't know how deal this such strange integral. and this problem…
math110
  • 93,304
16
votes
7 answers

How to calculate average of sine squared

I have problem with calculating average of sine. In my book, there is claim, that $$\langle\sin^2\omega t\rangle=\frac12$$ Now I have a question how to get this result mathematically? I know you must integrate by cycle, but I can not get to this…
user41502
16
votes
2 answers

Integral involving $\coth (x)$: Maple and Mathematica disagree

I was posed an interesting integral. $$ \int_{-\infty}^{\infty}\left(\frac{\coth(x)}{x^{3}}-\frac{1}{3x^{2}}-\frac{1}{x^{4}}\right)dx .$$ The integral evaluates to $\displaystyle\frac{-2}{{\pi}^{2}}\zeta(3)$ (Mathematica confirms this), but for…
Cody
  • 14,054
16
votes
1 answer

Does $f^3$ integrable imply $f$ integrable?

Let $f$ be a function defined on the close interval $[a,b]$. Does the riemann stieltjes integrability of $f^3$ imply the riemann stieltjes integrability of $f$ ? The answer is trivially no in the case of $f^2$, but I am not able to find a…
AlmostSureUser
  • 1,286
1
2
3
…
99 100