Question

If $f\left(x\right)=\left|\begin{array}{ccc}\cos x& e^{x^2}& 2x{\cos }^2\frac{x}{2}\\ x^2& \sec x& \sin x+x^3\\ 1& 2& x+\tan x\end{array}\right|$, then the value of ${\int }_{-\frac{\pi }{2}}^{\frac{\pi }{2}}(x^2+1)\left(f\right(x)+f^{\prime \prime }(x\left)\right)dx$, is equal to

• A. 1
• B. $-1$
• C. 2
• D. none of these

Answer 1

The correct option is D none of these

Clearly $f\left(x\right)$ is an odd function as, $f(-x)=-f\left(x\right)$
So $f^{\prime \prime }\left(x\right)$ will also be odd function and $x^2+1$ is an even function.
Hence required integral will vanish.