Question

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

Answer 1

`1/3(1 + e^(2x))^(3/2) + C`

Explanation:

Let `u = 1 + e^(2x)`. Then `du = 2e^(2x)dx` and `dx = (du)/(2e^(2x))`.

`int e^(2x)sqrt(u) * (du)/(2e^(2x))`

`1/2int sqrt(u) du`

`1/2(2/3u^(3/2)) + C`

`1/3u^(3/2) + C`

`1/3(1 + e^(2x))^(3/2) + C`

Hopefully this helps!