Question

How do you integrate `f(x)=3^(5x-1)/e^(4x)` using the quotient rule?

Answer 1

`int 3^(5x-1)/e^(4x) dx = 3^(5x-1)/(e^(4x)(5ln3-4))+C`

Explanation:

Note that:

`3^(5x-1)/e^(4x) = (e^(ln3))^(5x-1)*e^(-4x) = e^((5ln3 -4)x)*e^(-ln3) = 1/3 e^((5ln3 -4)x`

So:

`int 3^(5x-1)/e^(4x) dx = 1/3 int e^((5ln3 -4)x)dx = e^((5ln3 -4)x)/(3(5ln3-4))+C = 3^(5x-1)/(e^(4x)(5ln3-4))+C`