# How do you evaluate the integral int e^(2x)sqrt(1+e^(2x))?

Mar 1, 2017

$\frac{1}{3} {\left(1 + {e}^{2 x}\right)}^{\frac{3}{2}} + C$

#### Explanation:

Let $u = 1 + {e}^{2 x}$. Then $\mathrm{du} = 2 {e}^{2 x} \mathrm{dx}$ and $\mathrm{dx} = \frac{\mathrm{du}}{2 {e}^{2 x}}$.

$\int {e}^{2 x} \sqrt{u} \cdot \frac{\mathrm{du}}{2 {e}^{2 x}}$

$\frac{1}{2} \int \sqrt{u} \mathrm{du}$

$\frac{1}{2} \left(\frac{2}{3} {u}^{\frac{3}{2}}\right) + C$

$\frac{1}{3} {u}^{\frac{3}{2}} + C$

$\frac{1}{3} {\left(1 + {e}^{2 x}\right)}^{\frac{3}{2}} + C$

Hopefully this helps!