Evaluate∫10(x2−1)logxdx
Context:
I came across a similar integral some time ago, and now I would like to know how to tackle this. Any hint is very welcome.
Answer
Let f(k)=∫10xk−1logxdx
Then f′(k)=∫10xklogxlogxdx=∫10xkdx=11+k⟺f(k)−f(0)=log(1+k)⟺f(k)=log(1+k)
Evaluating k=2 we get f(2)=∫10x2−1logxdx=log(1+2)=log3
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