Question

integral de x senx/2 dx

Answer 1

$I=\int x\sin\left(\dfrac{x}{2}\right)dx\\\\ \text{Seja}~y=\dfrac{x}{2}\to x=2y\to dx=2dy.\text{ Substituindo acima:}\\\\ I=\int 2y\sin(y)~2dy=4\underbrace{\int y\sin(y)~dy}_{I_2}\\\\\\ \text{Usando que:}~\int udv=uv-\int vdu+C:\\\\ u=y\to du=dy\\\\ dv=\sin(y)dy\to v=-\cos(y)\\\\\\ \Longrightarrow I_2=-ycos(y)-\int(-\cos(y))dy\\\\ I_2=-y\cos(y)+\int \cos(y)dy\\\\ I_2=-y\cos(y)+\sin(y)$

$\text{Substituindo na express\~ao de }I:\\\\ I=4I_2\Longrightarrow\\\\I=4(-y\cos(y)+\sin(y))\\\\ I=4(-\dfrac{x}{2}\cos(\frac{x}{2})+\sin(\frac{x}{2}))\\\\ \boxed{\int x\sin\left(\dfrac{x}{2}\right)dx=4\sin\left(\dfrac{x}{2}\right)-2x\cos\left(\dfrac{x}{2}\right)+C}$