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

May 10, 2018

$\int \setminus \frac{1}{\sqrt{- {e}^{2 x} - 12 {e}^{x} - 37}} \setminus \mathrm{dx} = \frac{i}{\sqrt{37}} \setminus \text{arcsinh} \setminus \left(37 {e}^{- x} + 6\right) + C$

#### Explanation:

We seek:

$I = \int \setminus \frac{1}{\sqrt{- {e}^{2 x} - 12 {e}^{x} - 37}} \setminus \mathrm{dx}$

$\setminus \setminus = \int \setminus \frac{1}{\sqrt{\left(- 1\right) {e}^{2 x} \left(1 + 12 {e}^{- x} + 37 {e}^{- 2 x}\right)}} \setminus \mathrm{dx}$

$\setminus \setminus = \int \setminus \frac{1}{i {e}^{x} \sqrt{\left(1 + 12 {e}^{- x} + 37 {e}^{- 2 x}\right)}} \setminus \mathrm{dx}$

$\setminus \setminus = \int \setminus \frac{- i {e}^{- x}}{\sqrt{1 + 12 {e}^{- x} + 37 {e}^{- 2 x}}} \setminus \mathrm{dx}$

We can perform a substitution, Let:

$u = 37 {e}^{- x} + 6 \implies \frac{\mathrm{du}}{\mathrm{dx}} = - 37 {e}^{- x}$, and, ${e}^{- x} = \frac{u - 6}{37}$

The we can write the integral as:

$I = \int \setminus \frac{- \frac{i}{37}}{\sqrt{1 + 12 \left(\frac{u - 6}{37}\right) + 37 {\left(\frac{u - 6}{37}\right)}^{2}}} \setminus \mathrm{du}$

$\setminus \setminus = - \frac{i}{37} \setminus \int \setminus \frac{1}{\sqrt{1 + \frac{12}{37} \left(u - 6\right) + \frac{1}{37} {\left(u - 6\right)}^{2}}} \setminus \mathrm{du}$

$\setminus \setminus = - \frac{i}{37} \setminus \int \setminus \frac{1}{\sqrt{\frac{1}{37} \left(37 + 12 \left(u - 6\right) + {\left(u - 6\right)}^{2}\right)}} \setminus \mathrm{du}$

$\setminus \setminus = - \frac{i}{37} \setminus \int \setminus \frac{1}{\frac{1}{\sqrt{37}} \setminus \sqrt{37 + 12 u - 72 + {u}^{2} - 12 u + 36}} \setminus \mathrm{du}$

$\setminus \setminus = - \frac{i}{\sqrt{37}} \setminus \int \setminus \frac{1}{\sqrt{{u}^{2} + 1}} \setminus \mathrm{du}$

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

$I = - \frac{i}{\sqrt{37}} \setminus \text{arcsinh} \setminus u$

And if we restore the substitution, we get:

$I = - \frac{i}{\sqrt{37}} \setminus \text{arcsinh} \setminus \left(37 {e}^{- x} + 6\right) + C$