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

1 Answer
Mikee F.
Oct 6, 2015

#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) #

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