MHT CET · Maths · Definite Integration
The value of \(\int_0^2\left[x^2\right] \mathrm{d} x\) is (where \([x]\) denotes the greatest integer function not greater than \(x\) )
- A \(5-\sqrt{2}-\sqrt{3}\)
- B \(5+\sqrt{2}-\sqrt{3}\)
- C \(5+\sqrt{2}+\sqrt{3}\)
- D \(5-\sqrt{2}+\sqrt{3}\)
Answer & Solution
Correct Answer
(A) \(5-\sqrt{2}-\sqrt{3}\)
Step-by-step Solution
Detailed explanation
\(\int_0^2\left[x^2\right] \mathrm{d} x = \int_0^1 0 \, \mathrm{d} x + \int_1^{\sqrt{2}} 1 \, \mathrm{d} x + \int_{\sqrt{2}}^{\sqrt{3}} 2 \, \mathrm{d} x + \int_{\sqrt{3}}^{2} 3 \, \mathrm{d} x\) \(= 0 + [x]_1^{\sqrt{2}} + [2x]_{\sqrt{2}}^{\sqrt{3}} + [3x]_{\sqrt{3}}^{2}\)
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