∫ e x 1 + e x d x {\displaystyle \int e^{x}{\sqrt {1+e^{x}}}dx}
u = 1 + e x {\displaystyle u=1+e^{x}}
d u = e x d x {\displaystyle du=e^{x}dx}
∫ e x 1 + e x d x = ∫ u d u {\displaystyle \int e^{x}{\sqrt {1+e^{x}}}dx=\int {\sqrt {u}}du}
u d u = 2 u 2 / 3 3 + C {\displaystyle {\sqrt {u}}du={\frac {2u^{2/3}}{3}}+C}
= 2 3 ( e x + 1 ) 3 / 2 {\displaystyle ={\frac {2}{3}}(e^{x}+1)^{3/2}}
