Question

Evaluate the ∫e^z (sq rt of 1+e^z) dz

Answer 1

Solution.

$$\int e^z\sqrt{1+e^z}dz.$$

Make a replacement:

$t=\sqrt{1+e^z}, t\geq0,$ from here $t^2=1+e^z, \text{then } 2tdt=e^zdz.$

We will have

$\int t\cdot 2tdt=\int 2t^2dt=\newline= 2\frac{t^3}{3}+C= \frac{2}{3}(1+e^z)^{\frac{3}{2}}+C.$

Answer.

$$\frac{2}{3}(1+e^z)^{\frac{3}{2}}+C.$$