Question

If $f\left(x\right)=\left|\begin{array}{ccc}x& cosx& e^{\left|x\right|}\\ sinx& x^2& secx\\ tanx& 1& 2\end{array}\right|$ then the value of ${\int }_{-\pi /2}^{\pi /2}f\left(x\right)dx$ is equal to .................

Answer 1

$f\left(x\right)=\left|\begin{array}{ccc}x& cosx& e^{\left|x\right|}\\ sinx& x^2& secx\\ tanx& 1& 2\end{array}\right|f(-x)=\left|\begin{array}{ccc}-x& cos-x& e^{|-x|}\\ sin-x& {(-x)}^2& sec-x\\ tan-x& 1& 2\end{array}\right|=\left|\begin{array}{ccc}-x& cosx& e^{\left|x\right|}\\ -sinx& {\left(x\right)}^2& secx\\ -tanx& 1& 2\end{array}\right|=-\left|\begin{array}{ccc}x& cosx& e^{\left|x\right|}\\ sinx& x^2& secx\\ tanx& 1& 2\end{array}\right|$

As $f\left(x\right)+f(-x)=0$
Then ${\int }_{-\pi /2}^{\pi /2}f\left(x\right)=0$