Calculus of a Single Variable, Chapter 8, 8.6, Section 8.6, Problem 32

Indefinite integral follows the formula: int f(x) dx = F(x)+C
where:
f(x) as the integrand function
F(x) as the antiderivative of f(x)
C as constant of integration.
To evaluate the given integral problem: int (e^x)/(1-e^(2x))^(3/2) dx or int (e^xdx)/(1^2-(e^x)^2)^(3/2) , we may apply u-substitution by letting:
u =e^x then du = e^x dx .
Plug-in the values, the integral becomes:
int (e^xdx)/(1^2-(e^x)^2)^(3/2) =int (du)/(1^2-(u)^2)^(3/2)
In that form, it resembles one of the formulas from the integration table. It follows the integration formula for function with roots:
int dx/(a^2-x^2)^(3/2)= x/(a^2sqrt(a^2-x^2))+C
By comparing a^2 -x^2 and 1^2 -u^2 , we determine the corresponding values as: a=1 and x=u . Applying the integration formula, we get:
int (du)/(1^2-u^2)^(3/2) =u/(1^2sqrt(1^2-u^2))+C
=u/(1sqrt(1-u^2))+C
=u/sqrt(1-u^2)+C
Plug-in u =e^x on u/sqrt(1-u^2)+C , we get the indefinite integral as:
int (e^x)/(1-e^(2x))^(3/2) dx =(e^x)/sqrt(1-(e^x)^2)+C or (e^x)/sqrt(1-e^(2x))+C