Question

$\int\limits {\frac{mu+n}{1+u^2} } \, du$ Ajudem please.

Answer 1

$\displaystyle\mathsf{\int\dfrac{mu+n}{1+u^{2}}du } \\\displaystyle\mathsf{= \dfrac{1}{2} m\int\dfrac{2u}{1 + {u}^{2}}du + n \int \dfrac{du}{1 + {u}^{2} } }$

$\mathsf{t=1+u^{2}\to~dt=2u\,du}$

$\displaystyle\mathsf{\dfrac{1}{2}m\int\dfrac{2u} {1+u^{2}}du=\dfrac{1}{2}m \int \dfrac{dt}{t}} \\\mathsf{ \frac{1}{2}m \: \ell n |t| + k =\dfrac{1}{2}m \: \ell \:n |1 + {u}^{2} | + k }$

$\displaystyle\mathsf{n\int\dfrac{du}{1+u^{2}}=n\,arctg(u)+k}$

$\displaystyle\mathsf{\int\dfrac{mu+n}{1+u^{2}}du} \\\mathsf{=\dfrac{1}{2}m\,\ell\,n|1+u^{2}|+n\,arctg(u)+k}$