Integrate $\int_{0}^{n}{[2^x]}$, where $n$ is a natural number

Question

Source: MAT $2014$. Integrate the following - $$\int_{0}^{n}{[2^x]}$$ I thought of reversing $\frac{d}{dx}(a^x) = a^xln(a)$. So I divide and multiply by $ln(2)$. I end up with the final answer being $$\frac{2^n-1}{ln(2)}$$ This is not an answer in the multiple choices. I believe it's the floor function not allowing me to use the differentiation of $a^x$. How should I go about integrating a floor function?