Question

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

Answer 1

`int((1+e^2x)^(1/2))dx=2/(3e^2) (1+e^2x)^(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^2x`
(3) Take the derivative of both sides, hence, `2udu=e^2dx`
(4) Substitute 'u' and 'du' to the original differential eqn.
`int u * 2u/e^2 du`
(5) Integrate, `2/e^2intu^2du=2/(3e^2)u^3`
(6) Replace 'u' in terms of 'x' by using the defined value of 'u'
`int((1+e^2x)^(1/2))dx=2/(3e^2) (1+e^2x)^(3/2)`