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Q.
The value of the integral $\int\limits^{10}_{4} \frac{[x^2] dx}{[x^2 - 28x + 196] + [x^2]}$, where $[x]$ denotes the greatest integer less than or equal to $x$, is :
$I=\int\limits_{4}^{10} \frac{\left[x^{2}\right] d x}{\left[x^{2}-28 x+196\right]+\left[x^{2}\right]} \ldots \ldots .$ (i)
Use property $\int\limits_{a}^{b} f(a+b-x) d x=\int\limits_{a}^{b} f(x) d x$
$\Rightarrow I=\int\limits_{4}^{10} \frac{\left[x^{2}-28 x+196\right] d x}{\left[x^{2}\right]+\left[x^{2}-28 x+196\right]} \ldots \ldots .$ (ii)
by (i) and (ii)
$2 I=\int\limits_{4}^{10} d x=10-4=6$
$I =3$