Question

Assertion :If $\left[x\right]$ stands for greatest integer function, then the value of ${\int }_3^{15}\frac{\left[x^2\right]dx}{[x^2-36x+324]+\left[x^2\right]}$ is $6$ Reason: ${\int }_a^b\frac{f\left(x\right)}{f\left(x\right)+f(a+b-x)}dx=\frac{b-a}{2}$

• A. Both Assertion and Reason are correct and Reason is the correct explanation for Assertion
• B. Both Assertion and Reason are correct but Reason is not the correct explanation for Assertion
• C. Assertion is correct but Reason is incorrect
• D. Both Assertion and Reason are incorrect

Answer 1

The correct option is A Both Assertion and Reason are correct and Reason is the correct explanation for Assertion

Let $I={\int }_3^{15}\frac{\left[x^2\right]dx}{\left[{(x-18)}^2\right]+\left[x^2\right]}$ (i)
$I={\int }_3^{15}\frac{\left[{(18-x)}^2\right]dx}{\left[x^2\right]+\left[{(18-x)}^2\right]}$ (ii)
$(\because {\int }_a^{b}f\left(x\right)dx={\int }_a^{b}f(a+b-x)dx)$
adding (i) and (ii)

$2I={\int }_3^{15}1.dx=12$ $\therefore$ $I=6=\frac{15-3}{2}$
Hence , A