Question

How do you integrate `int 1/sqrt(e^(2x)+12e^x-45)dx` using trigonometric substitution?

Answer 1

`int frac{dx}{sqrt{e^{2x}+12e^x-45}} = frac{sqrt5*arcsin(frac{2}{3}-5e^{-x})}{15}+C`,
where `C` is the constant of integration.

Explanation:

Completing the square at the denominator gives

`e^{2x} + 12e^x - 45 -= (e^x + 6)^2 - 81`

To make use of the identity

`sec^2u - 1 -= tan^2u`,

substitute `9secu = e^x + 6`.

Answer 2

nice.. ...and the mighty Pythagorean right triangle supports this....

From this tiny little triangle you can glean TONS of valuable information and insight. Simple geometries can typically yield a beautiful perspective of things. On your own and using this triangle as your reference, find:
`1) tan(theta)`
`2) sec(theta)`
`3) sin(theta)`
`4) x`
`5) dx/(d theta)`
`6) dx`
`7) sqrt(e^(2x)+12e^x-45)`

No need to memorize formulas......that is WAY too much work and expended effort!!

Explanation:

Ok it work my bad