Question

Qual o valor da integral indefinida???

Answer 1

$\displaystyle\int\!x\,\mathrm{sen\,}5x\,dx$


Método de integração por partes:

$\begin{array}{lcl} u=x&~\Rightarrow~&du=dx\\\\ dv=\mathrm{sen\,}5x\,dx&~\Leftarrow~&v=-\,\dfrac{1}{5}\cos 5x \end{array}\\\\\\\\ \displaystyle\int\!u\,dv = uv-\int\!v\,du\\\\\\ \int\!x\,\mathrm{sen\,}5x\,dx =-\,\dfrac{1}{5}\,x\cos 5x-\int\!\left(-\,\dfrac{1}{5}\cos 5x \right )dx\\\\\\ \int\!x\,\mathrm{sen\,}5x\,dx =-\,\dfrac{1}{5}\,x\cos5x+\dfrac{1}{5}\int\!\cos 5x\,dx\\\\\\ \int\!x\,\mathrm{sen\,}5x\,dx =-\,\dfrac{1}{5}\,x\cos5x+\,\dfrac{1}{5}\cdot \left(\dfrac{1}{5}\,\mathrm{sen\,}5x \right )+C$

$\therefore~~\boxed{\begin{array}{c}\displaystyle\int\!x\,\mathrm{sen\,}5x\,dx =-\,\dfrac{1}{5}\,x\cos5x+\,\dfrac{1}{25}\,\mathrm{sen\,}5x+C \end{array}}$


Bons estudos! :-)