Integral Method Demystified
Let \(f: \mathbb{R}^{+} \rightarrow \mathbb{R}^{+} \), \(S ::= \sum\limits_{i=1}^{n} f(i) \), \(I ::= \int\limits_{1}^{n} f(x) dx \).

What is the upper bound for \(S \) when \(f \) is weakly increasing?

What is the upper bound for \(S \) when \(f \) is weakly decreasing?

What is the lower bound for \(S \) when \(f \) is weakly increasing?

What is the lower bound for \(S \) when \(f \) is weakly decreasing?

Do the upper bounds and lower bounds for \(f \) change if it is strictly increasing/decreasing instead of weakly increasing/decreasing?
From weakly increasing/decreasing to strictly increasing/decreasing, simply change the inequality sign from \(\le \) to <. 
What is the upper bound for \(H_n \)?

What is the lower bound for \(H_n \)?

What is the asymptotic bound for \(H_n \)?
The integral method is for finding the upper and lower bounds of a sum, but it is also helpful in obtaining the asymptotic bound / asymptotic equivalence / asymptotic equality of the sum. Once we have the upper and lower bounds, we take the limit on these bounds as \(n \rightarrow \infty \).