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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Two curious "identities" on $x^x$, $e$, and $\pi$

A numerical calculation on Mathematica shows that $$I_1=\int_0^1 x^x(1-x)^{1-x}\sin\pi x\,\mathrm dx\approx0.355822$$ and $$I_2=\int_0^1 x^{-x}(1-x)^{x-1}\sin\pi x\,\mathrm dx\approx1.15573$$ A furthur investigation on OEIS (A019632 and A061382)…
zy_
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What is the integral of 1/x?

What is the integral of $\frac{1}{x}$? Do you get $\ln(x)$ or $\ln|x|$? In general, does integrating $f'(x)/f(x)$ give $\ln(f(x))$ or $\ln|f(x)|$? Also, what is the derivative of $|f(x)|$? Is it $f'(x)$ or $|f'(x)|$?
hollow7
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Interpretation of an equality: Area of regular polygon and the integral $\int_0^{\infty}\! \frac{\mathbb{d}x}{1+x^N}$

Recently, I learned the integral from this post: $$\mathcal I=\int_0^{\infty}\! \frac{\mathbb{d}x}{1+x^N}=\frac{\pi/N}{\sin(\pi/N)}$$ This reminds me of the area of a regular polygon. Consider a $2N$-gon with unit "radius". (Distance from centre to…
Mythomorphic
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Evaluate $\int_0^{\pi/2}\log\cos(x)\,\mathrm{d}x$

How can you evaluate $$\int\limits_0^{\pi/2}\log\cos(x)\,\mathrm{d}x\;?$$
sujan
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How to prove that $\int_0^1\left(\sum\limits_{k=n}^\infty {x^k\over k}\right)^2\,dx = \int_0^1 2x^{n-1}\log\left(1+{1\over\sqrt{x}}\right)\,dx$

American Mathematical Monthly problem 11611 essentially asks you to show that $$\lim_n\ n \int_0^1\left(\sum_{k=n}^\infty {x^k\over k}\right)^2\,dx=2\log(2).\tag1$$ This would follow easily from (2) below, which is true for small values of $n$ …
user940
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Why does $\arctan(x)= \frac{1}{2i}\log \left( \frac{x-i}{x+i}\right)+k$?

Letting $x=\tan(u)$, $$\int\frac{1}{1+x^2} \, dx=\int\frac{1}{1+\tan(u)^2}\sec^2(u) \, du=u+k=\arctan(x)+k$$ Also, $$\int\frac{1}{1+x^2} \, dx=\int\frac{1}{(x-i)(x+i)} \, dx=\frac{1}{2i}\int\frac{1}{(x-i)}-\frac{1}{(x+i)} \,…
Meow
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interesting square of log sin integral

I ran across this challenging log sin integral and am wondering what may be a good approach. $$ \int_{0}^{\frac{\pi}{2}}x^{2}\ln^{2}(2\cos(x))dx=\frac{11{{\pi}^{5}}}{1440} $$ This looks like it may be able to be connected to the digamma or…
Cody
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Setting Up an Integral to Find A Cone's Surface Area

I tried proving the formula presented here by integrating the circumferences of cross-sections of a right circular cone: $$\int_{0}^{h}2\pi sdt, \qquad\qquad s = \frac{r}{h}t$$ so $$\int_{0}^{h}2\pi \frac{r}{h}tdt.$$ Integrating it got me $\pi h r$,…
G.P. Burdell
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Multivariate gaussian integral over positive reals

The multivariate gaussian integral over the whole $\mathbf{R}^n$ has closed form solution $$P = \int_{\mathbf{x} \in \mathbf{R}^n} \exp \left(-\frac12 \mathbf{x}^T \mathbf{A} \mathbf{x}\right)\,d\mathbf{x} = \sqrt{\frac{(2\pi)^n}{\det…
le_m
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Question from MIT integration Bee 2023 final: Evaluate $\int^1_0 (\sum^\infty_{n=0}\frac{\left\lfloor 2^nx\right\rfloor}{3^n})^2{\rm d}x$

I am trying to evaluate the last question from MIT integration Bee 2023 Final. $$\int^1_0 \left (\sum^\infty_{n=0}\frac{\left\lfloor 2^nx\right\rfloor}{3^n} \right )^2{\rm d}x$$ My approach is to divide $(0,1)$ into $1/2^n$ intervals and write the…
HeyFan
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Evaluating $\int_{-1}^{1}\frac{\arctan{x}}{1+x}\ln{\left(\frac{1+x^2}{2}\right)}dx$

This is a nice problem. I am trying to use nice methods to solve this integral, But I failed. $$\int_{-1}^{1}\dfrac{\arctan{x}}{1+x}\ln{\left(\dfrac{1+x^2}{2}\right)}dx, $$ where $\arctan{x}=\tan^{-1}{x}$ mark: this integral is my favorite one.…
math110
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Evaluating :$\int \frac{1}{x^{10} + x}dx$

$$\int \frac{1}{x^{10} + x}dx$$ My solution…
Frank
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Integral $\int_0^\infty\frac{dx}{\sqrt{1+\exp\left(\frac\pi2\left(x^2-\frac1{x^2}\right)\right) }}=\sqrt{\frac\pi2}$

Someone posted the integral on a local chat group $$ \int_0^\infty\frac{dx}{\sqrt{1+\exp\left(\dfrac\pi2\left(x^2-\dfrac1{x^2}\right)\right) }}=\sqrt{\frac\pi2} $$ It is interesting that the integrand is quite messy but the result is neat. Using…
Covariant
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Integrate $\int\sqrt\frac{\sin(x-a)}{\sin(x+a)}dx$

Integrate $$I=\int\sqrt\frac{\sin(x-a)}{\sin(x+a)}dx$$ Let $$\begin{align}u^2=\frac{\sin(x-a)}{\sin(x+a)}\implies…
RE60K
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Why doesn't integrating the area of the square give the volume of the cube?

I had a calculus course this semester in which I was taught that the integration of the area gives the size (volume): $$V = \int\limits_a^b {A(x)dx}$$ But this doesn't seem to work with the square. Since the size of the area of the square is $x^2$…
user54251
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