Question

como resolver essa integral: x sen(6x) dx?

Answer 1

$\int{x\mathrm{\,sen}(6x)\,dx}$


Utilizando o método de integração por partes

$\begin{array}{ll} u=x\;\;&\;\;du=dx\\ \\ dv=\mathrm{sen}(6x)\,dx\;\;&\;\;v=-\dfrac{1}{6}\cos (6x) \end{array}\\ \\ \\ \int{u\,dv}=uv-\int{v\,du}\\ \\ \\ \int{x\mathrm{\,sen}(6x)\,dx}=-\dfrac{1}{6}\,x\cos (6x)-\int{-\dfrac{1}{6}\cos (6x)\,dx}\\ \\ \\ \int{x\mathrm{\,sen}(6x)\,dx}=-\dfrac{1}{6}\,x\cos (6x)+\dfrac{1}{6}\int{\cos (6x)\,dx}\\ \\ \\ \int{x\mathrm{\,sen}(6x)\,dx}=-\dfrac{1}{6}\,x\cos (6x)+\dfrac{1}{6}\cdot \left[\dfrac{1}{6}\mathrm{\,sen}(6x) \right ]+C\\ \\ \\ \int{x\mathrm{\,sen}(6x)\,dx}=-\dfrac{1}{6}\,x\cos (6x)+\dfrac{1}{36}\mathrm{\,sen}(6x)+C$