Question

qual a integral de x sen (5x) dx

Answer 1

Calcular a integral indefinida:

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


Usaremos o método de integração por partes:

     $\begin{array}{lcl}u=x&\quad\Rightarrow\quad&du=dx\\\\\\ &&\displaystyle v=\int\mathrm{sen}(5x)\,dx\\\\ dv=\mathrm{sen}(5x)\,dx&\quad\Leftarrow\quad& v=-\,\dfrac{1}{5}\cos(5x) \end{array}$


     $\displaystyle\int u\,dv=uv-\int v\,du\\\\\\ \int x\,\mathrm{sen}(5x)\,dx=x\cdot \left(-\,\frac{1}{5}\cos(5x)\right)-\int \left(-\,\frac{1}{5}\cos(5x)\right)dx\\\\\\ \int x\,\mathrm{sen}(5x)\,dx=-\,\frac{1}{5}\,x\cos(5x)+\frac{1}{5}\int \cos(5x)\,dx\\\\\\ \int x\,\mathrm{sen}(5x)\,dx=-\,\frac{1}{5}\,x\cos(5x)+\frac{1}{5}\cdot \left(\frac{1}{5}\,\mathrm {sen}(5x)\right)+C$

     $\displaystyle\int x\,\mathrm{sen}(5x)\,dx=-\,\frac{1}{5}\,x\cos(5x)+\frac{1}{25}\,\mathrm {sen}(5x)+C\quad\longleftarrow\quad\textsf{resposta.}$


Bons estudos! :-)