Question

qual a integral √1+e^x dx?

Answer 1

$\displaystyle I=\int \sqrt{1+e^x}\,dx\\\\ u=(1+e^x)^{1/2}\to du = \dfrac{e^x}{2\sqrt{1+e^x}} dx \to dx =\dfrac{2u }{u^2-1}du\\ \\ I=\int \dfrac{2u^2}{u^2-1}\,du\\ \\ I=\int 2+\dfrac{2}{u^2-1}\,du\\ \\ I=2u + \int \dfrac{1}{u-1}-\dfrac{1}{u+1}\, du\\ \\ I=2u + \ln \left|\dfrac{u-1}{u+1}\right|+C\\ \\ \\ \boxed{I=2\sqrt{1+e^x}+\ln \left|\dfrac{\sqrt{1+e^x}-1}{\sqrt{1+e^x}+1}\right|+C}$