# How do you find the integral of (1 + e^2x) ^(1/2)?

Oct 6, 2015

$\int \left({\left(1 + {e}^{2} x\right)}^{\frac{1}{2}}\right) \mathrm{dx} = \frac{2}{3 {e}^{2}} {\left(1 + {e}^{2} x\right)}^{\frac{3}{2}}$

#### Explanation:

Integrate by method of substitution.

Solution:
(1) Let u = sqrt(1+e^2x
(2) Take the square of u, hence, ${u}^{2} = 1 + {e}^{2} x$
(3) Take the derivative of both sides, hence, $2 u \mathrm{du} = {e}^{2} \mathrm{dx}$
(4) Substitute 'u' and 'du' to the original differential eqn.
$\int u \cdot 2 \frac{u}{e} ^ 2 \mathrm{du}$
(5) Integrate, $\frac{2}{e} ^ 2 \int {u}^{2} \mathrm{du} = \frac{2}{3 {e}^{2}} {u}^{3}$
(6) Replace 'u' in terms of 'x' by using the defined value of 'u'
$\int \left({\left(1 + {e}^{2} x\right)}^{\frac{1}{2}}\right) \mathrm{dx} = \frac{2}{3 {e}^{2}} {\left(1 + {e}^{2} x\right)}^{\frac{3}{2}}$