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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Is this integral correct $\int_{-\pi/2}^{\pi/2}\cos^2(x)\sin[\alpha+\beta\tan( x)]\mathrm dx?$

How can we show that: $$\int_{-\pi/2}^{\pi/2}\cos^2(x)\sin[\alpha+\beta\tan( x)]\mathrm dx=\frac{1+\beta}{2}\cdot\frac{\pi}{e^{\beta}}\cdot\sin(\alpha)$$ assume $\alpha$ and $\beta$ are real numbers. I am not sure how to begin to tackle this…
user516887
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Evaluating an integral as the limit of a sum

Question: For $n ≥ 1$, let $G_n$ be the geometric mean of $\{\sin(\frac{\pi}{2}\frac{k}{n}): 1\le k \le n\} $. Then find $\lim_{n\to\infty}G_n $ My Approach: $G_n = \{\prod_{k=1}^n \sin(\frac{\pi}{2}\frac{k}{n})\}^{1/n}$ $\Rightarrow \ln G_n =…
Tamojit Maiti
  • 408
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Double integral with strange Change of Variables

I am trying to compute the following integral, $$\iint_{\mathbb{R}^2} \left(\frac{1-e^{-xy}}{xy}\right)^2 e^{-x^2-y^2}dxdy$$ First I tried substituting $x=r\cos{\theta}, y=r\sin{\theta}$ but it didn't really give me anything. For the second try, I…
zxcvber
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Why the two integrals are equal?

The equality $$\int_{0}^{2\pi}\left(-\frac{1+e^{-i\theta}}{1+\rho^{2}e^{i\theta}}\right)^{\!n}d\theta=\int_{\phi_{\rho}}^{2\pi-\phi_{\rho}}e^{in t}\frac{\sin(t/2)}{\sqrt{\rho^{2}-\cos^{2}(t/2)}}\,dt, $$ where $\phi_{\rho}=2\arccos(\rho)$, should…
Twi
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How to do this interesting integration?

$$\lim_{\Delta x\rightarrow0}\sum_{k=1}^{n-1}\int_{k+\Delta x}^{k+1-\Delta x}x^m dx$$ How to integrate the above integral? Edit1: $$\lim_{\Delta x\rightarrow0}\int_{2-\Delta x}^{2+\Delta x}x^m dx$$ Does this intergral give…
Inquisitive
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When $ f(x) = \int{f(x)} $

If $ f(x) = \int_{-\infty}^x{f(t) d t} $ means $ f(x) = B e^x $ Then $ f(x) = \int_0^x{f(t) d t} $ means $ f(x)=? $ EDIT: The obvious extension: $ f_a(x) = \int_a^x{f_a(t) d t} $ means $ f_a(x)=? $
Mark Hurd
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Long Integration Question

Integrate $$2\pi \int e^{-x}\sqrt{1+e^{-2x}} .dx $$ By looking at the question i knew it would lead to alot of calculations and working out but i've tried sooo many different methods and just end up in a mess.
jill
  • 121
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Is the integral $\int^{π/2}_0 \frac{\sin x}{x} \,\mathrm{d}x$ improper and why?

I am unable to understand the following question. Is the integral $$\int^\frac{π}{2}_0 \frac{\sin x}{x} \,\mathrm{d}x$$ improper and why? Also what is an impromper integral?
Farouq
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Clarification about conditions for using Feynman's Integration Technique

For the trick to work, we need to make sure beforehand that Leibniz's integral rule is satisfied, that is, given \begin{equation} \int_{a(x)}^{b(x)}f(x,t)dt \end{equation} we need that both $f(x,t)$ and it's partial in $x$ are continuous in $x$ and…
Paul Cusson
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Show that $ \int_{0}^{\infty}\left| \frac{\sin x}{x }\right| dx = ∞ $

How we can show that $$ \int_{0}^{∞} \frac {|\sin x|}{ |x| } dx = \infty? $$ I was thinking about the squeeze theorem as $$ \int_{0}^{\infty}\frac{1}{x}\, dx =\log x- \log0$$ where $x= \infty$. I don't know the proper proof, though.
user396850
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value of $f'(\sqrt π)+g'(\sqrt π)?$

Let $$f(x)=\left(\int_{0}^{x} e^{-t^2}dt\right)^2$$ and $$g(x)=\int_{0}^{1} \frac{e^{-x^2(1+t^2)}} {1+t^2} dt$$ Then what is the value of$$f'(\sqrt π)+g'(\sqrt π)?$$ I don't know how to solve this. But I guess in $g(x)$ we need to use gamma…
ASHWINI SANKHE
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Integral of $\,\arcsin(x)/(x^2)$

I did the integration by parts and got this expression, but then I am stuck on how to take it further. I tried substituting u=1-x^2, but then I had to do a partial fraction decomposition (which I did not take). Any hints or help would be very…
Hatem Chalak
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Indefinite integral involving arctan

How should I evaluate this definite integral? I am unable to figure out how to start. $$\int \tan^{-1} \left(1 + x + x^{2}\right) dx $$
Masked Man
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Simple integration question.

integrate $$ \int \sin(x) \cos(x)\; dx $$ using $u$-substitution. If i take $u = \sin(x)$ I get final answer to be $\sin^2(x) / 2 + c$ But If i take $u = \cos(x)$ I get final answer to be $-\cos^2(x) / 2 + c$ Are they equal? They should be,…
MathMathMath
  • 121
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Calculation $\int_0^1\frac{dx}{\sqrt{(1-x^2)(1-x^2k^2)}}$

My teacher said that calculate the following integral, and that the integral is convergent, I tried to calculate but failed, thanx in advanced for any help. $$0
Micheal Brain Hurts
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