Question

How do you find `\int e ^ { 2x } \sqrt { 1+ e ^ { 2x } } d x`?

Answer 1

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

Explanation:

The question asks for `int\ e^(2x)sqrt(1+e^(2x))\ dx`.

Substitute `u=1+e^(2x)`. Then, `(du)/dx=2e^(2x)`, or (solving for `dx`) `dx=(du)/(2e^(2x))`.

Then, `int\ e^(2x)sqrt(1+e^(2x))\ dx=int\ e^(2x)sqrt(u)\ (du)/(2e^(2x))`.

Simplify to get `1/2int\ sqrt(u)\ du=1/2int\ u^(1/2)\ du`. This is a simple integral. Use the power rule `int\ x^n\ dx=x^(n+1)/(n+1)+C`.

Then, `1/2int\ u^(1/2)\ du=(u^(3/2))/3+C=(sqrt(u^3))/3+C`. We want our answer to be in terms of `x`. Previously, `u` was defined as `1+e^(2x)`. Just substitute this in to get `(sqrt((e^(2x)+1)^3))/3+C`.