Integrate a^x

To integrate a^x, we first recognise that this is an exponential to the base a, often called a number to the power x.

When we see integration problems like these, the first thing to come to mind should be the base change method where a^x is the same as e^z, and we are changing the base from "a" to "e".

We take logs to the base e on both sides.

We find that ln e=1, and this simplifies it greatly.

This gives us an expression for z.

We substitute the expression for z into expression [1]

Hence, these two integrals are the same.

We integrate the easier version, which gives us the answer above.

We substitute expression [2] to remove e^(x lna).

This final answer is usually the standard formula given in some exams however; it is worth understanding the precise mechanism so you can use it for any base.

Examples