Question

How do you integrate `int e^x/sqrt(-e^(2x)-20e^x-96)dx` using trigonometric substitution?

Answer 1

`sin^(-1)((e^x+10)/2)+const.`

Explanation:

Considering the expression under the radical sign

`-(e^x+10)=-e^(2x)-20e^x-100`
`-> -e^(2x)-20e^x-96=2^2-(e^x+10)`

Now we can see a convenient trigonometric substitution such as:

`e^x+10=2siny`
`-> e^x*dx=2cosy*dy`

`=> int e^x/sqrt(-e^(2x)-20e^x-96)dx=int (2cos y)/sqrt(4-(2siny)^2)dy`
`=int (cancel(2)cosy)/(cancel(2)sqrt(1-sin^2y))dy=int cancel(cosy)/cancel(cosy)dy=int dy=y+const.`

But

`2siny=e^x+10` => `siny=(e^x+10)/2` => `y=sin^(-1)((e^x+10)/2)`

Then the result is
`sin^(-1)((e^x+10)/2) +const.`