Question

If $f\left(x\right)=\left|\begin{array}{ccc}2x{\sin }^2\left(\frac{x}{2}\right)& \tan x+x^3& x+\sin x\\ e^{x^2}& \sec x& 2\\ \cos x& x^4& 1\end{array}\right|$
Then value of $\underset{-\pi /2}{\overset{\pi /2}{\int }}\frac{\left[f\right(x)+f^{\prime \prime }(x\left)\right]}{2}dx=$

• A. $1$
• B. $-1$
• C. $2$
• D. None of the above

Answer 1

The correct option is D None of the above

$f\left(x\right)=\left|\begin{array}{ccc}2x{\sin }^2\left(\frac{x}{2}\right)& \tan x+x^3& x+\sin x\\ e^{x^2}& \sec x& 2\\ \cos x& x^4& 1\end{array}\right|$
$\Rightarrow f(-x)=-f\left(x\right)$
$\therefore f\left(x\right)$ is an odd function
$f^{\prime }\left(x\right)$ is an even funciton
$f^{\prime \prime }\left(x\right)$ is an odd funciton
$\Rightarrow \underset{-\pi /2}{\overset{\pi /2}{\int }}\frac{\left[f\right(x)+f^{\prime \prime }(x\left)\right]}{2}dx=0$