Question

Calcule a integral por partes

Answer 1

Olá!
  
    Sejam   $u=x,\;\; dv=e^{5x}$  . Temos:

$\displaystyle\int{u\;dv}=uv-\displaystyle\int{v\;du} \\ \\ \displaystyle\int{x\cdot e^{5x}}=x\cdot \dfrac{e^{5x}}{5}-\displaystyle \int{\dfrac{e^{5x}}{5}\cdot 1}dx=x\cdot \dfrac{e^{5x}}{5}-\dfrac{1}{5}\cdot \dfrac{e^{5x}}{5}+k,\;\; k\in\mathbb{R} = \\ \\= \dfrac{e^{5x}}{5}\cdot \left(x-\dfrac{1}{5}\right)+k,\;\; k\in\mathbb{R}$

Bons estudos!

Answer 2

$\sf \displaystyle \int \:x\cdot e^{5x}~dx\\\\\\=\int \frac{e^uu}{25}du\\\\\\=\frac{1}{25}\cdot \int \:e^uudu\\\\\\=\frac{1}{25}\left(e^uu-\int \:e^udu\right)\\\\\\=\frac{1}{25}\left(e^uu-e^u\right)\\\\\\=\frac{1}{25}\left(e^{5x}\cdot \:5x-e^{5x}\right)\\\\\\\to \boxed{\sf =\frac{1}{25}\left(e^{5x}\cdot \:5x-e^{5x}\right)+C}$