Question

How do you integrate `int 1/sqrt(-e^(2x)-12e^x-37)dx`?

Answer 1

`int \ 1/sqrt(-e^(2x)-12e^x-37) \ dx = i/sqrt(37) \ "arcsinh" \ (37e^(-x)+6) + C`

Explanation:

We seek:

`I = int \ 1/sqrt(-e^(2x)-12e^x-37) \ dx`

`\ \ = int \ 1/sqrt((-1)e^(2x)(1+12e^(-x)+37e^(-2x))) \ dx`

`\ \ = int \ 1/(ie^xsqrt((1+12e^(-x)+37e^(-2x)))) \ dx`

`\ \ = int \ (-ie^(-x))/(sqrt(1+12e^(-x)+37e^(-2x))) \ dx`

We can perform a substitution, Let:

`u = 37e^(-x)+6 => (du)/dx = -37e^(-x)`, and, `e^(-x)=(u-6)/37`

The we can write the integral as:

`I = int \ (-i/37)/(sqrt(1+12((u-6)/37)+37((u-6)/37)^2)) \ du`

`\ \ = -i/37 \ int \ (1)/(sqrt(1 + 12/37(u-6) + 1/37(u-6)^2)) \ du`

`\ \ = -i/37 \ int \ (1)/(sqrt(1/37(37 + 12(u-6) + (u-6)^2))) \ du`

`\ \ = -i/37 \ int \ (1)/(1/sqrt(37) \ sqrt(37 + 12u-72 + u^2-12u+36)) \ du`

`\ \ = -i/sqrt(37) \ int \ 1/sqrt(u^2+1) \ du`

This is a standard integral, so we can write (temporarily omitting the constant of integration):

`I = -i/sqrt(37) \ "arcsinh" \ u`

And if we restore the substitution, we get:

`I = -i/sqrt(37) \ "arcsinh" \ (37e^(-x)+6) + C`