Question

∫x²ln(x) DX

Integração por parte! Gabarito: 1/3x³ln(x)-x³/9

Answer 1

∫ x² ln (x)  = ∫ u dv = u·v - ∫ v du

u = ln (x)    du = 1/x dx
dv = x² dx   v= x³/3 

∫ x² ln (x) = 1/3 x³ ln(x) - 1/3 ∫ x³·1/x dx
∫ x² ln (x) = 1/3 x³ ln(x) - 1/3 ∫ x² dx
∫ x² ln (x) = 1/3 x³ ln(x) - x³/9 + c

Answer 2

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$\displaystyle\sf{\int x^2\ell n(x)~dx}\\\sf{fac_{\!\!,}a~u=\ell n(x)\implies du=\dfrac{1}{x}~dx}\\\sf{dv=x^2~dx\implies v=\dfrac{1}{3}x^3}\\\displaystyle\sf{\int x^2\ell n(x)~dx= \ell n(x)\cdot\dfrac{1}{3}x^3-\int \dfrac{1}{3}\diagup\!\!\!x^3\cdot\dfrac{1}{\diagup\!\!\!x}~dx}\\\displaystyle\sf{\int x^2\ell n(x)~dx=\dfrac{1}{3}x^3\ell n(x)-\dfrac{1}{3}\int x^2~dx}\\\displaystyle\sf{\int x^2\ell n(x)~dx=\dfrac{1}{3}x^3\ell n(x)-\dfrac{1}{3}\cdot\dfrac{1}{3}x^3+k}$

$\displaystyle\sf{\int x^2 \ell n(x)~dx=\dfrac{1}{3}x^3\ell n(x)-\dfrac{1}{9}x^3+k\checkmark}$