The value of the integral $I = \int_0^\pi [\sin x + \cos x](\cos x - \sin x) dx$ is equal to (where $[.]$ denotes the greatest integer function)
Step-by-Step Solution
Key Concept: Clever substitution $\sin x + \cos x = t$ transforms the denominator into a recognizable form
Let $\sin x + \cos x = t$, so $(\cos x - \sin x)dx = dt$. The integral becomes $I = \int_1^{\sqrt{2}} \frac{1}{t^2 - 1} dt = \sqrt{2} - 1$. This uses the substitution technique and standard integral formulas for rational functions.
Correct Answer: $\sqrt{2} - 1$