# A integration trick

I occasionally found an interesting trick about integration in Quora. It worth to be recorded.

Suppose you have a definite integral to calculate: $\int^1_0 \frac{x^3-1}{ln{x}} dx$, which is really hard to do.

Why not thinking about this function: $f(p) = \int^1_0 \frac{x^p-1}{\ln{x}} dx$ and we can differentiate $f(p)$ with respect to p.

Then, one could get: $f'(p) = \int^1_0 \frac{x^p\ln{x} }{\ln{x}} dx = \int^1_0 x^p dx = \frac{1}{p+1}$

So, one can easily integrate $f'(p)$ to get $f(p) = \ln{(p+1)} + \text{const}$.

Moreover, since $f(0)=\int^1_0 0 dx = 0$, we can pin down $\text{const}=0$\$ and finally got: $f(p) = \ln{(p+1)}$.

And the original integral is just: $\int^1_0 \frac{x^3-1}{\ln{x}} dx = f(3) = \ln{4}$