Let [ x ] denote the greatest integer x . Consider the function f ( x ) = max { x 2 , ? 1 + [ x ] } . Then the value of the integral 0 2 f ( x ) ? d x is [2023]

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Q.
Let $[x]$ denote the greatest integer $\leq x$ . Consider the function $f (x) = \max {x^{2}, 1 + [x]}$ . Then the value of the integral $\int_{0}^{2} f (x) d x$ is [2023]

1 $\frac{5 + 4 \sqrt{2}}{3}$

2 $\frac{8 + 4 \sqrt{2}}{3}$

3 $\frac{1 + 5 \sqrt{2}}{3}$

4 $\frac{4 + 5 \sqrt{2}}{3}$

Ans.
(1)

Given $f (x) = \max {x^{2}, 1 + [x]}$

So, $\int_{0}^{2} f (x) d x = \int_{0}^{1} 1 d x + \int_{1}^{\sqrt{2}} 2 d x + \int_{\sqrt{2}}^{2} x^{2} d x$

$= 1 + 2 (\sqrt{2} - 1) + \frac{8 - 2 \sqrt{2}}{3} = \frac{5 + 4 \sqrt{2}}{3}$

Solution

Q.
Let $[x]$ denote the greatest integer $\leq x$ . Consider the function $f (x) = \max {x^{2}, 1 + [x]}$ . Then the value of the integral $\int_{0}^{2} f (x) d x$ is [2023]

1 $\frac{5 + 4 \sqrt{2}}{3}$

2 $\frac{8 + 4 \sqrt{2}}{3}$

3 $\frac{1 + 5 \sqrt{2}}{3}$

4 $\frac{4 + 5 \sqrt{2}}{3}$

Ans.
(1)

Given $f (x) = \max {x^{2}, 1 + [x]}$

So, $\int_{0}^{2} f (x) d x = \int_{0}^{1} 1 d x + \int_{1}^{\sqrt{2}} 2 d x + \int_{\sqrt{2}}^{2} x^{2} d x$

$= 1 + 2 (\sqrt{2} - 1) + \frac{8 - 2 \sqrt{2}}{3} = \frac{5 + 4 \sqrt{2}}{3}$

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Solution

Q. Let [x] denote the greatest integer ≤x. Consider the function f(x)=max{x2, 1+[x]}. Then the value of the integral ∫02f(x) dx is [2023]

1 5+423

2 8+423

3 1+523

4 4+523

Ans. (1) Given f(x)=max{x2,1+[x]} So, ∫02f(x) dx=∫011 dx+∫122 dx+∫22x2 dx =1+2(2-1)+8-223=5+423

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Q.
Let $[x]$ denote the greatest integer $\leq x$ . Consider the function $f (x) = \max {x^{2}, 1 + [x]}$ . Then the value of the integral $\int_{0}^{2} f (x) d x$ is [2023]

1 $\frac{5 + 4 \sqrt{2}}{3}$

2 $\frac{8 + 4 \sqrt{2}}{3}$

3 $\frac{1 + 5 \sqrt{2}}{3}$

4 $\frac{4 + 5 \sqrt{2}}{3}$

Ans.
(1)

Given $f (x) = \max {x^{2}, 1 + [x]}$

So, $\int_{0}^{2} f (x) d x = \int_{0}^{1} 1 d x + \int_{1}^{\sqrt{2}} 2 d x + \int_{\sqrt{2}}^{2} x^{2} d x$

$= 1 + 2 (\sqrt{2} - 1) + \frac{8 - 2 \sqrt{2}}{3} = \frac{5 + 4 \sqrt{2}}{3}$