Question

$f\left(x\right)=\left|\begin{array}{ccc}\cos x& x& 1\\ 2\sin x& x^2& 2x\\ \tan x& x& 1\end{array}\right|$. If the value of $\underset{0}{\overset{\pi /4}{\int }}(\frac{f^{\prime }\left(x\right)}{x}-\frac{f\left(x\right)}{x^2})dx$ is ​$A$, then the value of $\left[A\right]$, where $\left[\right]$ denotes the greatest integer function, is​

Answer 1

$\underset{0}{\overset{\pi /4}{\int }}(\frac{f^{\prime }\left(x\right)}{x}-\frac{f\left(x\right)}{x^2})dx={\left[\frac{f\left(x\right)}{x}\right]}_0^{\pi /4}\frac{f\left(x\right)}{x}=\left|\begin{array}{ccc}\cos x& 1& 1\\ 2\sin x& x& 2x\\ \tan x& 1& 1\end{array}\right|$

$C_2\to C_2-C_3$
$\frac{f\left(x\right)}{x}=\left|\begin{array}{ccc}\cos x& 0& 1\\ 2\sin x& -x& 2x\\ \tan x& 0& 1\end{array}\right|$

$\Rightarrow \frac{f\left(x\right)}{x}=x(\tan x-\cos x)$

$\therefore A=\frac{\pi }{4}(1-\frac{1}{\sqrt{2}})\Rightarrow \left[A\right]=0$