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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Need help in evaluating $\int_{-1}^1 (1-x^2)^k, k \in \mathbb{N}$

Can someone tell me how to evaluate this integral please? $$\int_{-1}^1 (1-x^2)^k, k \in \mathbb{N}$$ I tried using the substitution x = sin(t), which would allow me to express this as: $$\int_{-1}^1 cos^{2k+1}(t) dt$$ but this doesn't really help.…
eager2learn
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High school integration problem

I love doing mathématique which are a little bit hard for my level to challenge myselft and push my mathématiques boundaries. And I come across a problem I can't understand how to finish. I have to prove that : $$\left |f(x) - \frac{1}{2x} \right…
pdubs
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Evaluate the following integral, $\int\sqrt{4-\sqrt{x}}dx$

Evaluate the following integral, $$\int\sqrt{4-\sqrt{x}}dx$$ $$\int \sqrt{4-\sqrt{x}}dx=\int \sqrt{2^2-(x^{1/4})^2}dx$$ Considering the common subsitution for $a^2-x^2$, let $$x^{1/4}=2\sin t$$ $$x=16\sin^4t$$$$\int dx=\int 64\sin^3t\cos t…
Yellow Skies
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Can we change $\triangle$ratio of Riemann integral? $\int_a^b f(x)(a(dx))^{b(dx)}\quad a,b\in \mathbb R^{\neq0}$

I wonder whether we can take integral as following or not? And, do they make any sense? $$f:\quad\text{is continious in [a,b]}$$$$\displaystyle\int_a^b f(x)(dx)^2\tag1$$ $$\displaystyle\int_a^b f(x)a^{(dx)}\quad\quad (a\in\mathbb…
Micheal Brain Hurts
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Is there a general formula for the integral $I_{n} = \int_{0}^{\frac{\pi}{2}} \sin^{2n-1}x + \sin^{2n-3}x + ... + \sin x dx, n\in \mathbb{N}$.

$$I_{n} = \int_{0}^{\frac{\pi}{2}} \sin^{2n-1}x + \sin^{2n-3}x + ... + \sin x dx, n\in \mathbb{N}$$ I'm using this integral to form part of answer to someone's question, but I'm struggling to find a formula for this integral (if one exists). Edit: I…
mrnovice
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Solve $ \frac{2}{\pi}\int_{-\pi}^\pi\frac{\sin\frac{9x}{2}}{\sin\frac{x}{2}}dx $

Solve the following integral: $$ \frac{2}{\pi}\int_{-\pi}^\pi\frac{\sin\frac{9x}{2}}{\sin\frac{x}{2}}dx $$
user417036
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Evaluate $ \int \frac{1}{\sin{x}\cos^{3}x}dx $

$$ \int \frac{1}{\sin{x}\cos^{3}x}dx $$ $$\Rightarrow \int \frac{1}{\sin{x}\cos^{3}x}{\cos{x}\over \cos{x}}dx$$ $$\Rightarrow \int \frac{\sec^{4}{x}}{\tan{x}}dx$$ $$\Rightarrow \int \frac{\sec^{2}(1+\tan^{2}x)}{\tan{x}}dx$$ $$Substitution \tan{x}=t…
Raknos13
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How to deal with the integral of $e^x e^{x^2}$?

I'm struggling with the integral $$\int e^x e^{x^2} \mathrm{d}x$$ how can you possibly integrate that?
user43783
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How to evaluate $\int_{0}^{1}\frac{\log x}{1+x} dx$

Today When I was computing measure theoretic entropy of Gauss map I encountered this integral. Then I check asked Questions but I couldn't find same question how to evaluate: $$\int_{0}^{1}\frac{\log x}{1+x} dx$$ Thanks for any hint.
M.H
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Integral of $\int\frac{dx}{x^2\sqrt{1-\frac{1}{x^2}}}$

I was asked to find the following integral: $$\int\frac{dx}{x^2\sqrt{1-\frac{1}{x^2}}}$$ What I tried to replace $\sqrt{1-\frac{1}{x^2}}$ with $u$ so that: $$du=\frac{dx}{x^3\sqrt{1-\frac{1}{x^2}}} \Rightarrow…
Jonathan Martinez
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Integral of $\sec^2{x} \tan^2{x}$

I'm once again stuck; I'm trying to find $$\int{(\sec^2{x} \tan^2{x})dx}$$ but end up with things like: $\tan^2x-\int{2\tan^2x \sec^2x}$, which doesn't help. Would it be best to approach using integration by parts or substitution?
Tobi
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How to integrate $\int \frac{(ar)}{\sqrt{a^2 - r^2}} dr $ ?

How to integrate $\int \frac{(ar)}{\sqrt{a^2 - r^2}} dr $ ? I tried making $ u = a^2 - r^2 $ but I can't seem to get $ -a\sqrt{a^2 - r^2} $ Any help is appreciated! Thank you.
user403648
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Alternative method of solving $\int_0^{\pi/2} {\sin^2{x} \ln{\tan x} \,dx}$

Solve the integral: $$\int_0^{\pi/2} {\sin^2{x} \ln{\tan x} \,dx}$$ I have already found the answer to be $\frac{\pi}{4}$ by the method explained below, but I would like to know whether there is another way. --- My method --- Use a $u$-sub: $u=\tan…
Ant
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Feynman integration - help with this old answer

I need help understanding an answer: https://math.stackexchange.com/a/1808872/335418 . The answer is from @Quantum spaghettification (who from his profile stats seems to be no longer active on this site): I asked for clarification there, but it…
Srini
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What are the differences among the different definitions of an integral?

For a non-mathematician (physicist) all the integrals and definitions are equal so what are the differences among: The Lebesgue integral The Riemann Integral The Riemann-Stiejles integral Why aren't all the same?
Jose Garcia
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