WBJEE · Maths · Definite Integration
If \([x]\) denotes the greatest integer less than or equal to \(x\), then the value of the integral \(\int_{0}^{2} x^{2}[x] d x\) equals
- A \(\frac{5}{3}\)
- B \(\frac{7}{3}\)
- C \(\frac{8}{3}\)
- D \(\frac{4}{3}\)
Answer & Solution
Correct Answer
(B) \(\frac{7}{3}\)
Step-by-step Solution
Detailed explanation
Let. \[ \begin{aligned} I &=\int_{0}^{2} x^{2}[x] d x \\ &=\int_{0}^{1} x^{2} \cdot 0 d x+\int_{1}^{2} x^{2} \times 1 d x \\ &=\left[\frac{x^{3}}{3}\right]_{1}^{2}=\left[\frac{8}{3}-\frac{1}{3}\right]=\frac{7}{3} \end{aligned} \]
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