JEE MAIN - Mathematics (2016 - 10th April Morning Slot - No. 18)
The value of the integral
$$\int\limits_4^{10} {{{\left[ {{x^2}} \right]dx} \over {\left[ {{x^2} - 28x + 196} \right] + \left[ {{x^2}} \right]}}} ,$$
where [x] denotes the greatest integer less than or equal to x, is :
$$\int\limits_4^{10} {{{\left[ {{x^2}} \right]dx} \over {\left[ {{x^2} - 28x + 196} \right] + \left[ {{x^2}} \right]}}} ,$$
where [x] denotes the greatest integer less than or equal to x, is :
6
3
7
$${1 \over 3}$$
Explanation
Let I = $$\int\limits_4^{10} {{{\left[ {{x^2}} \right]\,dx} \over {\left[ {{x^2} - 28x + 196} \right] + \left[ {{x^2}} \right]}}} $$
= $$\int\limits_4^{10} {{{\left[ {{x^2}} \right]dx} \over {\left[ {{{\left( {x - 14} \right)}^2}} \right] + \left[ {{x^2}} \right]}}} \,\,\,.....(1)$$
Using,
$$\int\limits_a^b {f\left( {a + b - x} \right)dx\,} $$ = $$\,\,\int\limits_a^b {f(x)\,\,dx} $$
I = $$\int\limits_4^{10} {{{{{\left( {x - 14} \right)}^2}} \over {\left[ {{x^2}} \right] + \left[ {{{\left( {x - 14} \right)}^2}} \right]}}} \,dx\,\,....(2)$$
Adding (1) and (2)
2I = $$\int\limits_4^{10} {{{\left[ {{{\left( {x - 14} \right)}^2}} \right] + \left[ {{x^2}} \right]} \over {\left[ {{x^2}} \right] + \left[ {{{\left( {x - 14} \right)}^2}} \right]}}} \,dx$$
$$ \Rightarrow $$$$\,\,\,$$ 2I = $$\int\limits_4^{10} {dx} = \left[ x \right]_4^{10}$$ = 6
= I = 3
= $$\int\limits_4^{10} {{{\left[ {{x^2}} \right]dx} \over {\left[ {{{\left( {x - 14} \right)}^2}} \right] + \left[ {{x^2}} \right]}}} \,\,\,.....(1)$$
Using,
$$\int\limits_a^b {f\left( {a + b - x} \right)dx\,} $$ = $$\,\,\int\limits_a^b {f(x)\,\,dx} $$
I = $$\int\limits_4^{10} {{{{{\left( {x - 14} \right)}^2}} \over {\left[ {{x^2}} \right] + \left[ {{{\left( {x - 14} \right)}^2}} \right]}}} \,dx\,\,....(2)$$
Adding (1) and (2)
2I = $$\int\limits_4^{10} {{{\left[ {{{\left( {x - 14} \right)}^2}} \right] + \left[ {{x^2}} \right]} \over {\left[ {{x^2}} \right] + \left[ {{{\left( {x - 14} \right)}^2}} \right]}}} \,dx$$
$$ \Rightarrow $$$$\,\,\,$$ 2I = $$\int\limits_4^{10} {dx} = \left[ x \right]_4^{10}$$ = 6
= I = 3
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