Usando o método de integração por partes: ∫ u .dv = u .v- ∫ v.du, e lembrando que (ln x)’ =1/x,determine I= ∫ ln x.5x^4 dx
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![\displaystyle \int (lnx.5x^{4})dx\\\\\
u=lnx~~~~du= (\frac{1}{x})dx \\\\\
dv=5 x^{4} ~~~~~v= \frac{5 x^{5} }{5} ~~~~v= x^{5}](https://tex.z-dn.net/?f=%5Cdisplaystyle+%5Cint+%28lnx.5x%5E%7B4%7D%29dx%5C%5C%5C%5C%5C%0Au%3Dlnx%7E%7E%7E%7Edu%3D+%28%5Cfrac%7B1%7D%7Bx%7D%29dx+%5C%5C%5C%5C%5C%0Adv%3D5+x%5E%7B4%7D+%7E%7E%7E%7E%7Ev%3D+%5Cfrac%7B5+x%5E%7B5%7D+%7D%7B5%7D+%7E%7E%7E%7Ev%3D+x%5E%7B5%7D+%0A "\displaystyle \int (lnx.5x^{4})dx\\\\\
u=lnx~~~~du= (\frac{1}{x})dx \\\\\
dv=5 x^{4} ~~~~~v= \frac{5 x^{5} }{5} ~~~~v= x^{5}")
![\displaystyle \int (lnx.5x^{4})=u.v-\displaystyle \int v.du\\\\\\\\\\
\displaystyle \int (lnx.5x^{4})=lnx. x^{5} -\displaystyle \int (x^{5} . \frac{1}{x}) dx\\\\\\\\\
\displaystyle \int (lnx.5x^{4})=lnx. x^{5} -\displaystyle \int ( \frac{ x^{5} }{x}) dx\\\\\\\\\
\displaystyle \int (lnx.5x^{4})=lnx. x^{5} -\displaystyle \int ( x^{4} ) dx\\\\\\\\\
\displaystyle \int (lnx.5x^{4})=lnx. x^{5} - \frac{ x^{5} }{5}+c](https://tex.z-dn.net/?f=%5Cdisplaystyle+%5Cint+%28lnx.5x%5E%7B4%7D%29%3Du.v-%5Cdisplaystyle+%5Cint+v.du%5C%5C%5C%5C%5C%5C%5C%5C%5C%5C%0A%5Cdisplaystyle+%5Cint+%28lnx.5x%5E%7B4%7D%29%3Dlnx.+x%5E%7B5%7D+-%5Cdisplaystyle+%5Cint++%28x%5E%7B5%7D+.+%5Cfrac%7B1%7D%7Bx%7D%29+dx%5C%5C%5C%5C%5C%5C%5C%5C%5C%0A%5Cdisplaystyle+%5Cint+%28lnx.5x%5E%7B4%7D%29%3Dlnx.+x%5E%7B5%7D+-%5Cdisplaystyle+%5Cint++%28+%5Cfrac%7B+x%5E%7B5%7D+%7D%7Bx%7D%29+dx%5C%5C%5C%5C%5C%5C%5C%5C%5C%0A%5Cdisplaystyle+%5Cint+%28lnx.5x%5E%7B4%7D%29%3Dlnx.+x%5E%7B5%7D+-%5Cdisplaystyle+%5Cint++%28++x%5E%7B4%7D+%29+dx%5C%5C%5C%5C%5C%5C%5C%5C%5C%0A%5Cdisplaystyle+%5Cint+%28lnx.5x%5E%7B4%7D%29%3Dlnx.+x%5E%7B5%7D+-+%5Cfrac%7B+x%5E%7B5%7D+%7D%7B5%7D%2Bc+%C2%A0 "\displaystyle \int (lnx.5x^{4})=u.v-\displaystyle \int v.du\\\\\\\\\\
\displaystyle \int (lnx.5x^{4})=lnx. x^{5} -\displaystyle \int (x^{5} . \frac{1}{x}) dx\\\\\\\\\
\displaystyle \int (lnx.5x^{4})=lnx. x^{5} -\displaystyle \int ( \frac{ x^{5} }{x}) dx\\\\\\\\\
\displaystyle \int (lnx.5x^{4})=lnx. x^{5} -\displaystyle \int ( x^{4} ) dx\\\\\\\\\
\displaystyle \int (lnx.5x^{4})=lnx. x^{5} - \frac{ x^{5} }{5}+c")
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