Resolver ∫ intxe^xz wrt x

$$\int \int x e ^ { x z } d x$$

Ebaluatu

$$\left\{\begin{matrix}\frac{xze^{xz}-2e^{xz}}{z^{3}}+С\left(x+1\right),&z\neq 0\\Сx+\frac{x^{3}}{6}+С_{1},&z=0\end{matrix}\right.$$

Ebazpidea

Sinplifikatu.

$$\left(\frac{-e^{xz}+xze^{xz}}{z^{2}}+С_{3}\right)x-\frac{x^{2}e^{xz}}{z}+\frac{2\left(-e^{xz}+xze^{xz}\right)}{z^{3}}$$

Integratu gehiketa gaiz gai.

$$\int \frac{x^{2}}{2}\mathrm{d}x+\int С_{4}\mathrm{d}x$$

Deskonposatu konstantea gaika.

$$\frac{\int x^{2}\mathrm{d}x}{2}+\int С_{4}\mathrm{d}x$$

Baldin k\neq -1rentzat \int x^{k}\mathrm{d}x=\frac{x^{k+1}}{k+1}, ordeztu \int x^{2}\mathrm{d}x \frac{x^{3}}{3}rekin. Egin \frac{1}{2} bider \frac{x^{3}}{3}.

$$\frac{x^{3}}{6}+\int С_{4}\mathrm{d}x$$

Aurkitu С_{4}en integrala integral arrunten taulako \int a\mathrm{d}x=ax araua erabiliz.

$$\frac{x^{3}}{6}+С_{4}x$$

F\left(x\right) f\left(x\right)ren jatorrizkoa bada, orduan f\left(x\right)ren jatorrizko guztien multzoa ematen du F\left(x\right)+Ck. Beraz, gehitu C\in \mathrm{R} integrazio-konstantea emaitzari.

$$\left\{\begin{matrix}\left(\frac{-e^{xz}+xze^{xz}}{z^{2}}+С_{3}\right)x-\frac{x^{2}e^{xz}}{z}+\frac{2\left(-e^{xz}+xze^{xz}\right)}{z^{3}}+С_{7},&\\\frac{x^{3}}{6}+С_{4}x+С_{7},&\end{matrix}\right.$$

Diferentziatu x balioarekiko

$$\left\{\begin{matrix}\frac{xze^{xz}-e^{xz}}{z^{2}}+С,&z\neq 0\\\frac{x^{2}}{2}+С,&z=0\end{matrix}\right.$$