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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Confusion in options of integration

If $$ I(a)=\int_0^\infty\frac{\arctan(ax)-\arctan(x)}{x}\,\mathrm dx, $$ then (A) $I'(1),I'(2),I'(3)$ are in harmonic progression. (B) $I'(2)=\dfrac\pi4$ (C) $I(\pi)=\dfrac\pi2\log\pi$ (D) $I'(3)=\dfrac\pi6$ In this question the derivative…
search
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Integral of $\cos(\sqrt{x^3})$

As the title explains, I can't seem to get anywhere on this one, so could someone show me how I would proceed with solving this? $$\int\cos\left(\sqrt{x^3}\right)\,dx$$ I can't shrug off the feeling that I'll end up somewhere in the complex realm.
DerpyMcDerp
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Faster way to compute the integral

Here it is: $$\int_o^\pi x\cos^4x\,dx$$ I used integration by parts but I would be grateful if someone told me an alternate method to compute the integral faster.
user43081
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Integrating $\frac{\log(1+x)}{1+x^2}$

Possible Duplicate: Evaluate the integral: $\int_{0}^{1} \frac{\ln(x+1)}{x^2+1} dx$ I am a bit stuck here in evaluating the following integral:$$\int_{0}^{1}\frac{\log(1+x)}{1+x^2}\,\mathrm dx$$.Your help is appreciated.
user43081
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Dimension analysis of an integral

I'm reading Street-Fighting Mathematics and not sure if I understand integral dimension analysis. The idea is to "guess" integrals without explicit calculation, by just looking at their dimensions. It's been a good decade since I last touched…
andreister
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finding $ \int^{\infty}_{0}\frac{\ln x}{x^2+6x+9}dx$

finding $\displaystyle \int^{\infty}_{0}\frac{\ln x}{x^2+6x+9}dx$ Attempt: let $\displaystyle I(a) = \int^{\infty}_{0}\frac{\ln (ax)}{(x+3)^2}dx, a>0$ $\displaystyle I'(a) = \int^{\infty}_{0}\frac{x}{ax(x+3)^2}dx =…
DXT
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Evaluate integral with integer part

I have to evaluate $$\int _0^2\:\frac{x-\left[x\right]}{2x-\left[x\right]+1}dx$$ Where $[x] = floor(x)$ I tend to write it like this, but I think i'm missing the point $x = 2$ $$\int _0^2\:\frac{x-\left[x\right]}{2x-\left[x\right]+1}dx=\int…
Liviu
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How to integrate this arcsin integral?

How to integrate $$\int_a^1 \frac{\arcsin x}{\sqrt{x^2-a^2}} dx?$$ This integral appeared in Demkov Yu. N., Ostrovsky V. N., Berezina (Avdonina) N. B., "Uniqueness of the Firsov Inversion Method and Focusing Potentials", Sov. Phys. JETP 33, 867-70,…
Sarah
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Integrating $\int \frac{\cos x}{\sin x+\cos x}dx$.

so I've just had my first exam (went pretty well) but I ran into this thing as the first part of the last question. $$\int \frac{\cos x}{\sin x+\cos x}dx$$ I had a look on wolfram after the exam and it advised to multiply top and bottom by…
Ryan Tandy
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Evaluate $\int x^2e^{\frac{x^2}2} \, dx$

$$\int x^2e^{\frac{x^2}2} \, dx$$ I do not need to evaluate $\int e^{\frac{x^2}{2}}$ in a numeric method. If we take $u=x^2, u'=2x$ $v'=e^{\frac{x^2}{2}},v=\int e^{\frac{x^2}2}\,dx$ we get: $$\int x^2e^{\frac{x^2}{2}} \, dx = x^2\int…
gbox
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How to integrate $\int\sqrt{\frac{4-x}{4+x}}$?

Let $$g(x)=\sqrt{\dfrac{4-x}{4+x}}.$$ I would like to find the primitive of $g(x)$, say $G(x)$. I did the following: first the domain of $g(x)$ is $D_g=(-4, 4]$. Second, we have \begin{align} G(x)=\int g(x)dx &=\int\sqrt{\dfrac{4-x}{4+x}}dx\\ …
Zir
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$\int f(x) dx\cdot \int \frac{1}{f(x)}dx = c$ where $c$ is a constant. Find $f(x)$

$$\int f(x) dx\cdot \int \frac{1}{f(x)}dx = c$$ where $c$ is a constant. Find $f(x)$. Off first glance it seems that $f(x)$ is some form of $e^x$, but how does one go about doing this analytically (only 1 mark for guessing). I took the…
bigfocalchord
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How to fix this stupid mistake while keeping it as simple as it is.

In this answer I used some symmetry to show that $$ \int_0^1 \frac{dx}{x+ \sqrt{1-x^2}} = \frac \pi 4. $$ Then I thought about whether I could make it simpler by avoiding trigonometric substitutions, like this: $$ y =…
Michael Hardy
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Is integrating $f(x)= x\exp(-x^2/2)$ with substitution $u = x^2$ well defined?

I have the following question, which is: I know how to integrate $f(x) = x\exp(-x^2)$ using the standard method of subsitution, however I was wondering if this method is well defined, and if so why exactly. The reason I ask this, is because a…
Frederick del Pozo
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How do you evaluate the integral $\int\frac{x^2-1}{(x^4+3 x^2+1) \tan^{-1}\left(\frac{x^2+1}{x}\right)}\,dx$?

i'm required to evaluate this integral. I've tried factorizing but it doesn't lead me to anywhere. $$\int\frac{x^2-1}{(x^4+3 x^2+1) \tan^{-1}\left(\frac{x^2+1}{x}\right)}\,dx$$ I've also tried letting $u = \frac{x^2+1}{x}$, $du/dx$ gets me…
jaclynx
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