Question

$Letf\left(x\right)=\left|\begin{array}{ccc}a\left(x\right)& b\left(x\right)& c\left(x\right)\\ m\left(x\right)& n\left(x\right)& l\left(x\right)\\ g\left(x\right)& h\left(x\right)& k\left(x\right)\end{array}\right|then{\int }_0^{a}f\left(x\right)is\left|\begin{array}{ccc}{\int }_0^{a}a\left(x\right)& {\int }_0^{a}b\left(x\right)& {\int }_0^{a}c\left(x\right)\\ {\int }_0^{a}m\left(x\right)& {\int }_0^{a}n\left(x\right)& {\int }_0^{a}l\left(x\right)\\ {\int }_0^{a}g\left(x\right)& {\int }_0^{a}h\left(x\right)& {\int }_0^{a}k\left(x\right)\end{array}\right|$

• A. F
• B. T

Answer 1

The correct option is A

F

Integration in determinants is not straight forward. But is is simple in some special cases like only one of the rows having elements as function of x and we will discuss only such cases in this standard.

Supposef(x)= $\left|\begin{array}{ccc}a\left(x\right)& b\left(x\right)& c\left(x\right)\\ 4& 3& 4\\ 1& 5& 3\end{array}\right|$ , then ${\int }_0^{a}f\left(x\right)$ will be
$\left|\begin{array}{ccc}{\int }_0^{a}a\left(x\right)& {\int }_0^{a}b\left(x\right)& {\int }_0^{a}c\left(x\right)\\ 4& 3& 7\\ 1& 5& 3\end{array}\right|$, but if f(x)=

$\left|\begin{array}{ccc}a\left(x\right)& b\left(x\right)& c\left(x\right)\\ m\left(x\right)& n\left(x\right)& l\left(x\right)\\ g\left(x\right)& h\left(x\right)& k\left(x\right)\end{array}\right|$, then ${\int }_0^{a}f\left(x\right)$ is not equal to

$\left|\begin{array}{ccc}{\int }_0^{a}a\left(x\right)& {\int }_0^{a}b\left(x\right)& {\int }_0^{a}c\left(x\right)\\ {\int }_0^{a}m\left(x\right)& {\int }_0^{a}n\left(x\right)& {\int }_0^{a}l\left(x\right)\\ {\int }_0^{a}g\left(x\right)& {\int }_0^{a}h\left(x\right)& {\int }_0^{a}k\left(x\right)\end{array}\right|$