Question

How do you integrate `int 1/sqrt(-e^(2x) +100)dx` using trigonometric substitution?

Answer 1

`1/10 int (dy)/cosy`

Explanation:

Making `e^{-x} = 10 cosy` we have

`- e^{-x}dx = -10 sin y dy`

and

`dx = 10 siny/e^{-x}dy = tan y dy`

so

`int (dx)/sqrt(-e^(2x) +100) equiv 1/10int (tany)/sinydy = 1/10 int (dy)/cosy`