Question

$f\left(x\right)=\left|\begin{array}{ccc}3& 1& 2\\ a^3& 2a& a^2\\ 3x^2& 2& 2x\end{array}\right|then{\int }_0^{a}f\left(x\right)is$ ___

Answer 1

There is only one row with functions in the given determinant expression. Therefore

${\int }_0^{a}f\left(x\right)=f\left(x\right)=\left|\begin{array}{ccc}3& 1& 2\\ a^3& 2a& a^2\\ {\int }_0^{a}3x^2& {\int }_0^{a}2& {\int }_0^{a}2x\end{array}\right|=\left|\begin{array}{ccc}3& 1& 2\\ a^3& 2a& a^2\\ (a^3-0)& (2a-0)& (a^2-0)\end{array}\right|=\left|\begin{array}{ccc}3& 1& 2\\ a^3& 2a& a^2\\ a^3& 2a& a^2\end{array}\right|=0.$