Taula da integraal

Integraal definiit

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

${[F(x)]}_a^b=\underset{a}{\overset{b}{\int }}f(x)dx=F(b)-F(a)$

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:

$\underset{0}{\overset{+\mathrm{\infty }}{\int }}\sqrt{x}{e}^{-x}dx=\frac{1}{2}\sqrt{\pi }$

$\underset{0}{\overset{+\mathrm{\infty }}{\int }}{e}^{-\frac{x^2}{2}}dx=\sqrt{\frac{\pi }{2}}$ (integrala da Gauss)

$\underset{0}{\overset{+\mathrm{\infty }}{\int }}{e}^{-x^2}dx=\frac{1}{2}\sqrt{\pi }$

$\underset{0}{\overset{+\mathrm{\infty }}{\int }}\frac{x}{e^x-1}dx=\frac{{\pi }^2}{6}$

$\underset{0}{\overset{+\mathrm{\infty }}{\int }}\frac{x^3}{e^x-1}dx=\frac{{\pi }^4}{15}$

$\underset{0}{\overset{+\mathrm{\infty }}{\int }}\frac{\sin (x)}{x}dx=\frac{\pi }{2}$

$\underset{0}{\overset{+\mathrm{\infty }}{\int }}{x}^{z-1}{e}^{-x}dx=\mathrm{\Gamma }(z)$ ($\mathrm{\Gamma }$ a l'è la funziun Gamma, definida pour z > 0)

$\underset{0}{\overset{1}{\int }}\frac{1}{\sqrt{1-t^3}}dt=\frac{1}{3}\mathrm{B}(\frac{1}{3},\frac{1}{2})$ (integrala elítica)

$\underset{0}{\overset{\frac{\pi }{2}}{\int }}\ln (\cos (x))dx=\underset{0}{\overset{\frac{\pi }{2}}{\int }}\ln (\sin (x))dx=-\frac{\pi }{2}\ln (2)$

$\underset{-\mathrm{\infty }}{\overset{+\mathrm{\infty }}{\int }}\cos (x^2)dx=\underset{-\mathrm{\infty }}{\overset{+\mathrm{\infty }}{\int }}\sin (x^2)dx=\sqrt{\frac{\pi }{2}}$ (integrall da Fresnel)

$\underset{0}{\overset{\pi }{\int }}\ln (1-2\alpha \cos x+{\alpha }^2)dx=2\pi \ln |\alpha |$ si $|\alpha |>1$ e $0$ si $|\alpha |\le 1$.

$\underset{0}{\overset{+\mathrm{\infty }}{\int }}x{e}^{-x^3}dx=\frac{1}{3}\mathrm{\Gamma }\left(\frac{2}{3}\right)$

$\underset{0}{\overset{\frac{\pi }{2}}{\int }}{\sin }^n(x)dx=I_n$ (integrall da Wallis)

Videe igualameent

• Taula da primitiif
• Integrala
• Integrala indefinida