Metoda supstitucije

Metoda supstitucije je metoda rešavanja integrala u kojoj se deo integrala zamenjuje jednostavnijim simbolom (obično se koristi latinično slovo u), u cilju da bi se dobio integral koji je lakše rešiti.^[1]

${\displaystyle \int {\frac {\ln {x}}{x}}dx=?}$

${\displaystyle u=ln(x)}$

${\displaystyle du={\frac {1}{x}}dx}$

${\displaystyle xdu=dx}$

${\displaystyle \int {\frac {ln(x)}{x}}dx=\int {\frac {u}{x}}xdu={\frac {u^{2}}{2}}+C}$

${\displaystyle \int {\frac {ln(x)}{x}}dx={\frac {(\ln(x))^{2}}{2}}+C}$

U nekim slučajevima moraće se rešiti za ${\displaystyle x}$

${\displaystyle \int 2x{\sqrt {5x-20}}dx=?}$

${\displaystyle \omega =5x-20}$

${\displaystyle d\omega =5dx}$

${\displaystyle {\frac {d\omega }{5}}=dx}$

${\displaystyle x={\frac {\omega +20}{5}}}$

${\displaystyle \int 2x{\sqrt {5x-20}}dx=\int 2\left({\frac {\omega +20}{5}}\right){\sqrt {\omega }}{\frac {d\omega }{5}}={\frac {2}{25}}\int (\omega +20){\sqrt {\omega }}d\omega ={\frac {2}{25}}\int (\omega {\sqrt {\omega }}+20{\sqrt {\omega }})d\omega ={\frac {2}{25}}\left(\int \omega {\sqrt {\omega }}d\omega +\int 20{\sqrt {\omega }}d\omega \right)}$

${\displaystyle {\frac {2}{25}}\left(\int \omega {\sqrt {\omega }}d\omega +\int 20{\sqrt {\omega }}d\omega \right)={\frac {2}{25}}\left({\frac {\omega ^{\frac {5}{2}}}{\frac {5}{2}}}+20{\frac {\omega ^{\frac {3}{2}}}{\frac {3}{2}}}\right)+C}$

${\displaystyle \int 2x{\sqrt {5x-20}}dx={\frac {2}{25}}\left({\frac {(5x-20)^{\frac {5}{2}}}{\frac {5}{2}}}+20{\frac {(5x-20)^{\frac {3}{2}}}{\frac {3}{2}}}\right)+C}$

Metoda supstitucije se može koristiti za definisanje novih antiderivata.

${\displaystyle \int \tan {x}dx=?}$

${\displaystyle \int \tan {x}dx=\int {\frac {\sin {x}}{\cos {x}}}dx}$

${\displaystyle u=\cos {x}}$

${\displaystyle du=-\sin {x}dx}$

${\displaystyle {\frac {du}{-sin{x}}}=dx}$

${\displaystyle \int {\frac {\sin {x}}{\cos {x}}}dx=\int {\frac {sin{x}}{u}}{\frac {du}{-\sin {x}}}=-\int {\frac {1}{u}}du=-\ln {u}+C}$

${\displaystyle \int \tan {x}dx=\ln {cos{x}}+C}$