How do you find the antiderivative of (e^(2x))(sqrt(1+3e^(2x)))(e2x)(√1+3e2x)?

1 Answer
Eddie
Aug 1, 2016

=1/9 (1 + 3 e^(2x))^(3/2) + C=19(1+3e2x)32+C

Explanation:

there's a simple pattern here

d/dx (1 + 3 e^(2x))^(3/2)ddx(1+3e2x)32

=3/2 (1 + 3 e^(2x))^(1/2) * 6 e^(2x)=32(1+3e2x)12⋅6e2x

=9 e^(2x) (1 + 3 e^(2x))^(1/2) =9e2x(1+3e2x)12

so d/dx( 1/9 (1 + 3 e^(2x))^(3/2) )= e^(2x) (1 + 3 e^(2x))^(1/2)ddx(19(1+3e2x)32)=e2x(1+3e2x)12

so

int \ e^(2x) (1 + 3 e^(2x))^(1/2) \ dx = int \ d/dx ( 1/9 (1 + 3 e^(2x))^(3/2) )\ dx

=1/9 (1 + 3 e^(2x))^(3/2) + C

of course you might wish to use a sub u = e^(2x) etc, but this way allows you to do it pretty much in your head.

Related questions
See all questions in Integrals of Exponential Functions
Impact of this question
1591 views around the world
You can reuse this answer
Creative Commons License