Question

Why is the integral of e^x * a^x equal to (e^x * a^x / 1 + ln a) + C ?

How did we get the + 1 in the denominator? I think it's just some basic rules but I still do not get it.

Answer 1

It follows from `a^x = (e^{ln a})^x = e^{(ln a)x}`

Explanation:

`e^xa^x = e^x e^{(ln a)x} = e^{(1+ln a)x}`

Since `int e^{alpha x}dx = 1/alpha e^{alpha x}+C`, we have

`int e^xa^x dx = int e^{(1+ln a)x} dx = e^{(1+ln a)x}/{1+ln a}+C`