The value of $\int_{1}^{2} \frac{x}{1+\lfloor x \rfloor} dx$ (where $[x]$ is greatest integer function) is
Step-by-Step Solution
Key Concept: Using trigonometric substitution and the relationship between sine and cosine to evaluate definite integrals.
$I = \int_0^\pi \frac{\sin^2 x}{1 + \sin x} dx = 4\int_0^\pi \sqrt{\sin x} \cos x dx$. Using substitution $u = \sin x$: $I = [\sin^2 x]_0^\pi = \frac{4}{3}$. Therefore, $|I| = 2$.
Correct Answer: 2