Question

Let [.] denote the greatest integer function then the value of $\int\limits^{1.5}_{0}x \left[x^{2}\right]$ dx is : .

• A. 0
• B. $\frac{3}{2}$
• C. $\frac{3}{4}$
• D. $\frac{5}{4}$

Answer 1

Answer: (C)

$\frac{3}{4}$

Answer 2

$\int\limits^{1}_{0} x\left[x^{2}\right]dx+\int\limits^{\sqrt{2}}_{1}x\left[x^{2}\right]dx+\int\limits^{1.5}_{\sqrt{2}}x\left[x^{2}\right]dx$ $\int\limits^{1}_{0}x.0dx +\int\limits^{\sqrt{2}}_{1}xdx +\int\limits^{1.5}_{\sqrt{2}}2x \,dx$ $0+\left[\frac{x^{2}}{2}\right]^{\sqrt{2}}_{1}+\left[x^{2}\right]^{1.5}_{\sqrt{2}}$ $\frac{1}{2}\left(2-1\right)+\left(2.25-2\right)$ $\frac{1}{2}+.25$ $\frac{1}{2}+\frac{1}{4} = \frac{3}{4}$