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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What is the closed form for $\int_{0}^{\infty}\frac{1}{1+x^2}\cdot\frac{x^{\pi}}{1+x^{\pi}}\cdot\frac{1}{1+x^ e}dx $?

On my previou page Jack D'Aurizio offered a concise elegant prove of Vladimir Reshetnikov's identity and a closed form for…
user335850
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Is the following integral positive?

For each positive integer $n$, consider the following intgral: $$\int_0^\infty\int_0^\infty\frac{(x^4-y^2)x^{2n+1}}{(x^4+y^2)[(1+x^2)^2+y^2]^{n+1}}dxdy.$$ I want to know if there is any easy way to see that it's positive. If there is no easy proof…
Paul
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How to perform abstract integration

Suppose $X$ is a Banach space. Here, page $124$, there is this sentence If $\mu$ is a finite support on $X$ we can define its barycenter $\beta(\mu)=\beta_X(\mu) \in X$ by $$\beta(\mu)=\int x d\mu$$ In the following Lemma $2.4$, it states that…
Idonknow
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Evaluate the integration : $\int\sqrt{\frac{(1-\sin x)(2-\sin x)}{(1+\sin x)(2+\sin x)}}dx$

$$\int{\sqrt{\frac{(1-\sin x)(2-\sin x)}{(1+\sin x)(2+\sin x)}}dx}$$ $$\int\sqrt{\frac{(1-\sin x)(2-\sin x)}{(1+\sin x)(2+\sin x)}}dx=\int \frac{(1-\sin x)(2-\sin x)}{\sqrt{(1-\sin x)(2-\sin x)(1+\sin x)(2+\sin x)}}dx$$ I am stuck. Please help…
Brahmagupta
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Integral depending on a parameter

Task: find all values of the parameter, such that integral converges. $$\int_0^{+\infty} \frac{dx}{1+x^a \sin^2x}$$ I tried a lot and I used Cauchy and Weierstrass method but it was useless. And now I know that I must use $$\sum_n \int_{\pi…
Simankov
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Prove that $\int^2_0 x(8-x^3)^\frac{1}{3}dx=\frac{16\pi}{9\sqrt{3}}$

I have to prove that: $$\int^2_0 x(8-x^3)^\frac{1}{3}dx=\frac{16\,\pi}{9\sqrt{3}}.$$ I tried substituting $x^3=8u$ but I just got stuck. Any help would be appreciated.
chriskinda
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A problem about parametric integral

How to solve the following integral. $I(\theta) = \int_0^{\pi}\ln(1+\theta \cos x)dx$ where $|\theta|<1$
BerSerK
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Any advice on simplifying this nasty integral

Can anyone think of any smart way of approximating this nasty integral $$F = \int_{-c}^c f_X(x) \, dx $$ where $c$ is a non-negative constant (for example $\frac{1}{64}$) and where the integrand is given by $$f_X(x)=…
Henry
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Integral $\int^{1}_{-1} \frac{\ln(ax^2+2bx+a)}{x^2+1}dx$ if $a>b>0$

I am trying to evaluate the following integral: $$\int^{1}_{-1} \frac{\ln(ax^2+2bx+a)}{x^2+1}dx,$$ where $a>b>0$. I can't really think of a way to find it. So, please give me a hint.
Eri
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Evaluating $\int_0^2(\tan^{-1}(\pi x)-\tan^{-1} x)\,\mathrm{d}x$

Hint given: Write the integrand as an integral. I'm supposed to do this as double integration. My attempt: $$\int_0^2 [\tan^{-1}y]^{\pi x}_{x}$$ $$= \int_0^2 \int_x^{\pi x} \frac { \mathrm{d}y \mathrm{d}x} {y^2+1}$$ $$= \int_2^{2\pi}…
Diya
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How find this integral $I=\int_{0}^{+\infty}\frac{x}{1+e^x}dx$

Question: $$I=\int_{0}^{+\infty}\dfrac{x}{1+e^x}dx$$ I know use $$I=\int_{0}^{\infty}\dfrac{xe^{-x}}{1+e^{-x}}dx=\sum_{k=0}^{\infty}\int_{0}^{\infty}xe^{-(k+1)x}dx=\sum_{k=1}^{\infty}(-1)^{k-1}\cdot\dfrac{1}{k^2}=\dfrac{\pi^2}{12}$$ someone have…
math110
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If $f$ is a Riemann integrable prove $|f|$ is also Riemann integrable

Show that if $f$ is Riemann integrable on $[a,b]$ then $|f|$ is also Riemann integrable on $[a,b]$. My idea is: let $f$ be in $[a,b]$ less than $|f|$, since $f$ is integrable then $|f|$ is also integrable on $[a,b]$.
nichodemus
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Integration of $\int (\frac{1-x}{1+x})^{\frac{1}{3}}$

I've been trying to integrate this for a long time but can't. $$\int \left(\frac{1-x}{1+x}\right)^{\frac{1}{3}}$$ I tried assuming $\frac{1-x}{1+x}=t^3$ , also tried integration by parts but it gets stuck in a loop.
user129048
  • 337
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How to solve the integral $\int \frac {(x^2 +1)}{x^4- x^2 +1} dx$

I have started this problem but I'm not completely sure I'm going down the right path with it. So far I have completed the square in the denominator. $x^4-x^2+1= (x^2-1/2)^2+\frac{3}{4}$ Then, let $u=x^2-\frac{1}{2}$ so…
user175401
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Prove that the integral is positive

I'm trying to prove that the integral: $$ \int_0^{2\pi} f(x)\cos(x)\, dx $$ is positive. It has continuous first and second derivatives and such that $f''(x)>0$ for $0
Overclock
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