Question

Sa se calculeze folosint integrarea pein parti:
integrala din( x^2 * lnx) dx

Answer 1

$\int {x}^{2}lnx\:dx$

$f=lnx=>f'=(lnx)'=\frac{1}{x}$

$g'={x}^{2}=>g=\int {x}^{2}\:dx=\frac{{x}^{2+1}}{2+1}=\frac{{x}^{3}}{3}$

$\int f \times g' \:dx=f \times g-\int f' \times g \:dx$

$=\frac{{x}^{3}}{3}lnx-\int \frac{1}{x} \times \frac{{x}^{3}}{3} dx$

$=\frac{{x}^{3}}{3}lnx-\frac{1}{3} \int \frac{{x}^{3}}{x} dx$

$=\frac{{x}^{3}}{3}lnx-\frac{1}{3} \int {x}^{2} dx$

$=\frac{{x}^{3}}{3}lnx-\frac{1}{3} \times \frac{{x}^{3}}{3}$

$=\frac{{x}^{3}}{3}lnx-\frac{{x}^{3}}{9}$

$=\frac{3{x}^{3}lnx-{x}^{3}}{9}$

$=\frac{{x}^{3}(3lnx-1)}{9}+C$