List of top Mathematics Questions on Definite Integrals asked in CPET

Evaluate: \(\displaystyle\int_0^2 \left([x^2]-[x]^2\right)dx\), where \([\cdot]\) denotes the greatest integer (floor) function.

Consider the improper integrals \(I_\lambda = \displaystyle\int_0^1 \dfrac{dx}{(1-x)^\lambda}\) and \(J_\lambda = \displaystyle\int_1^\infty \dfrac{dx}{x^\lambda}\), \(\lambda \in \mathbb{R}\), and the statements:
(I) For \(\lambda = 1\), \(I_\lambda\) converges, but \(J_\lambda\) diverges.
(II) For \(\lambda = 2\), \(I_\lambda\) diverges, but \(J_\lambda\) converges.
Choose the correct option.

The value of \(\displaystyle\int_0^{\pi/2} \sin^2(x)\cos^4(x)\,dx\) is ____.