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Integrate $$\int_0^1 \frac{x^7-1}{\log x} \, \mathrm{d}x$$!It's from MIT integration bee

I try to solve this by substitution but nothing works out. Solving this problem from 10 days .I have no idea how they have a answer $\log(8)$.

Ketan
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Niraj
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1 Answers1

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The trick we use is differentiating under the integral sign.

Define $$F(k) = \int_{0}^{1} \frac{x^k - 1}{\log(x)}dx$$

Then, $$\frac{\partial F(k) }{\partial k}= \int_{0}^{1} \frac{x^k \log(x) }{\log(x)}dx =\int_{0}^{1}x^k dx= \frac{1}{k+1}$$

Our original integral can be retrieved by integrating with respect to $k$ and setting $k=7$: $$F(7) = \int_{0}^{7}\frac{1}{k+1}dk = \log(8)$$

amWhy
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Race Bannon
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    Parametric integration really ought to be taught more widely in, say, second-semester calculus along with the usual "techniques of integration". It is simple enough conceptually and is just too darn useful to set aside "for later". – colormegone Jul 19 '15 at 16:44
  • What is the justification for interchanging the order of integration and differentiation here? – Calculon Jul 19 '15 at 16:47
  • See https://en.wikipedia.org/wiki/Differentiation_under_the_integral_sign for a formal proof and many examples – Race Bannon Jul 19 '15 at 16:49
  • A short sentence about the differentiation through the integral would have been nice. +1 anyway. – Nicolas Jul 19 '15 at 16:50
  • This reminds me of something that would be found in the "handbook of integration" a few mits putt outfit a few years ago. – JMJ May 06 '17 at 21:35
  • That should be "put out." – JMJ May 06 '17 at 21:36