Question

Qual a resolução da integral
(2x+1)senxdx

Answer 1

$I=\displaystyle\int{(2x+1)\,\mathrm{sen\,}x\,dx}$


Método de integração por partes:

$\begin{array}{lcl} u=2x+1&~\Rightarrow~&du=2\,dx\\ \\ dv=\mathrm{sen\,}x\,dx&~\Leftarrow~&v=-\cos x \end{array}\\ \\ \\ \\ \displaystyle\int{u\,dv}=uv-\int{v\,du}\\ \\ \\ \int{(2x+1)\,\mathrm{sen\,}x\,dx}=(2x+1)\cdot (-\cos x)-\int{(-\cos x)\cdot 2\,dx}\\ \\ \\ \int{(2x+1)\,\mathrm{sen\,}x\,dx}=-(2x+1)\cos x+2\int{\cos x\,dx}\\ \\ \\ \boxed{\begin{array}{c}\displaystyle\int{(2x+1)\,\mathrm{sen\,}x\,dx}=-(2x+1)\cos x+2\,\mathrm{sen\,}x+C \end{array}}$