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.

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