Clues on how to solve these types of problems within 2-3 minutes for competitive exams

$$\int_0^{102}\left(\prod_{k=1}^{100}(x-k)\right)\left(\sum_{k=1}^{100}\frac1{x-k}\right)\,dx$$

I've tried solving this problem but only thing that comes to my mind is the manual integration by multiplication of the expressions which will literally take much longer than the allotted time for competitive exams Now this is a homework and exercises problem but I'd be glad if I could get some clues on how to solve this problem.

Solution 1:

Hint:

By the product rule you have the following result:

$$\dfrac{\mathrm d}{\mathrm dx}\prod_{k=1}^{100}(x-k)=\left(\prod_{k=1}^{100}(x-k)\right)\left(\sum_{k=1}^{100}\dfrac{1}{(x-k)}\right)$$

Integrate both sides from $0$ to $102$, use the Fundamental Theorem of Calculus and you'll be done in no time.

Solution 2:

Here's a quick hint: if you differentiate the product in the integrand, you get the entire integrand so by the fundamental theorem of calculus you can evaluate this very fast.