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}

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