# How to solve this integral? This one is the hardest! [duplicate]

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 $$\log(8)$$.

• can you use $\LaTeX$ please! Commented Jul 19, 2015 at 16:25
• And what are the limits of integration? Commented Jul 19, 2015 at 16:26

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 $k=7$: $$F(7) = \int_{0}^{7}\frac{1}{k+1}dk = \log(8)$$

• 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". Commented Jul 19, 2015 at 16:44
• What is the justification for interchanging the order of integration and differentiation here? Commented Jul 19, 2015 at 16:47
• See en.wikipedia.org/wiki/Differentiation_under_the_integral_sign for a formal proof and many examples Commented Jul 19, 2015 at 16:49
• A short sentence about the differentiation through the integral would have been nice. +1 anyway. Commented Jul 19, 2015 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
Commented May 6, 2017 at 21:35