# Integral of a^{x}

How to solve the given integral by using the integral of a^{x}: formula, 1 example, and its solution.

## Formula

The derivative of [1/(ln a)]⋅a^{x} is

[1/(ln a)]⋅[a^{x} (ln a)] = a^{x}.

derivative of a^{x}

And integral and derivative

are the opposite operations.

So the integral of a^{x} is

[1/(ln a)]a^{x} + C.

## Example∫ 2^{x}(3^{x} - 1) dx

Solution

Solution (Detail)

2^{x}(3^{x} - 1) = 6^{x} - 2^{x}

The integral of 6^{x} is

[1/(ln 6)]⋅6^{x}.

Write the coefficient -.

The integral of 2^{x} is

[1/(ln 2)]⋅2^{x}.

The given integral is an indefinite integral.

So write +C.

So

[1/(ln 6)]⋅6^{x} - [1/(ln 2)]⋅2^{x} + C

is the answer.