Question

xln (x) dx como resol essa integral indefinida

Answer 1

Boa tarde Odilon!

Solução!

Você resolve essa integral por partes.

$\displaystyle \int xlnxdx$


$u=lnx~~~~dv=xdx\\\\\ du= \dfrac{1}{x}dx~~~~v= \dfrac{ x^{2} }{2}dx \\\\\\\\ \displaystyle \int xlnxdx=uv-\displaystyle \int vdu \\\\\\\\ \displaystyle \int xlnxdx=lnx.\dfrac{ x^{2} }{2}-\displaystyle \int\dfrac{ x^{2} }{2}.\dfrac{1}{x}dx$




$\displaystyle \int xlnxdx=lnx.\dfrac{ x^{2} }{2}- \dfrac{1}{2} \displaystyle \int\dfrac{ x^{2} }{x}dx\\\\\\\\\\\ \displaystyle \int xlnxdx=lnx.\dfrac{ x^{2} }{2}- \dfrac{1}{2} \displaystyle \int xdx\\\\\\\\\\\ \displaystyle \int xlnxdx=lnx.\dfrac{ x^{2} }{2}- \dfrac{1}{2} \frac{ x^{2} }{2}+c \\\\\\\\\\ \displaystyle \int xlnxdx=lnx.\dfrac{ x^{2} }{2}- \frac{ x^{2} }{4}+c \\\\\\\\\\\\\\ \boxed{Resp:~~\displaystyle \int xlnxdx=\dfrac{ lnx.x^{2} }{2}- \frac{ x^{2} }{4}+c}$


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Bons estudos!