Question

If $x_1$ and $x_2\left(x_1<x_2\right)$ are two values of $x$ satisfying the equation $â2\left(x^2+\frac{1}{x^2}\right)+ââ1−x^2â=4\left(\frac{3}{2}−2^{x^2−3}−\frac{1}{2^{x^2+1}}\right)$, then the value of $\underset{x_1+x_2}{\overset{3x_2−x_1}{∫}}\left\{\frac{x}{4}\right\}\left(1+\left[\tan \left(\frac{\{x\}}{1+\{x\}}\right)\right]\right)dx$, (where $[â‹\dots ]$ and $[â‹\dots ]$ denote greatest integer function and fractional part function respectively)

Answer 1

Answer: 2

$⇒x=Â\pm 1⇒x_1=−1$
$x_2=1$
$\underset{0}{\overset{4}{∫}}\frac{x}{4}1+\tan \frac{x}{1+x}dx$
$\underset{0}{\overset{4}{∫}}\frac{x}{4}dx=\frac{1}{4}\frac{x^2}{2}0$
$=\frac{1}{4}[8−0]=2$