Taula da integraal

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[redatá] Integraal definiit

Sa la cjama integrala definida da la funziun f in l'intervall [a,b] la grandezza

\left[ F(x) \right]_{a}^{b} = \int_{a}^{b} f(x)\,dx = F(b) - F(a)

intúe F\, l'è una primitiva qual-sa-vöör da f\, e a\,, b\, a inn i cunfin da l'integrala.

I esiist di funziun che a inn integràbil, ma da che vargüna primitiva la pöö mía vess esprimüda in tèrmin da funziun elementaar. Da tüta manera, la valuur da vargüna da sti integrall a pöö vess calcülada e a l'è mustrada chí-da-sota:

\int_0^{+\infty}{\sqrt{x}\,e^{-x}\,dx} = \frac{1}{2}\sqrt \pi
\int_0^{+\infty}{e^{-\frac{x^2}{2}}\,dx} = \sqrt{\frac{\pi}{2}} (integrala da Gauss)
\int_0^{+\infty}{e^{-x^2}\,dx} = \frac{1}{2}\sqrt \pi
\int_0^{+\infty}{\frac{x}{e^x-1}\,dx} = \frac{\pi^2}{6}
\int_0^{+\infty}{\frac{x^3}{e^x-1}\,dx} = \frac{\pi^4}{15}
\int_0^{+\infty}\frac{\sin(x)}{x}\,dx=\frac{\pi}{2}
\int_0^{+\infty}  x^{z-1}\,e^{-x}\,dx = \Gamma(z) (\Gamma\, a l'è la funziun Gamma, definida pour z > 0)
\int_0^1  \frac{1}{\sqrt{1-t^3}}\,dt = \frac{1}{3}\Beta\left(\frac{1}{3}, \frac{1}{2}\right) (integrala elítica)
\int_0^{\frac{\pi}{2}} \ln(\cos(x))\, dx=\int_0^{\frac{\pi}{2}} \ln(\sin(x))\, dx=-\frac{\pi}{2}\ln(2)
\int_{-\infty}^{+\infty}\cos(x^2)\,dx=\int_{-\infty}^{+\infty}\sin(x^2)\,dx= \sqrt{\frac{\pi}{2}} (integrall da Fresnel)
\int_0^{\pi} \ln(1-2\alpha\cos\,x+\alpha^2)\,dx= 2\pi\ln|\alpha| si |\alpha|>1\, e 0\, si |\alpha|\leq 1.
\int_0^{+\infty}{xe^{-x^3}\,dx} = \frac{1}{3}\Gamma\left(\frac{2}{3}\right)
\int_0^{\frac{\pi}{2}} \sin^n(x)\,dx = I_n (integrall da Wallis)

[redatá] Videe igualameent

  • Taula da primitiif
  • Integrala
  • Integrala indefinida