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

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.

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.