Question

If ${\int }_0^{3}f\left(x\right).dx={\int }_3^{0}f\left(x\right).dx$ then

• A. f(x) is an exponential function
• B. f(x) is a logarithmic function
• C. f(x) is a constant function
• D. f(x) = 0

Answer 1

The correct option is D f(x) = 0

As we have learnt one of the main property of Definite Integrals is if we reverse the upper and lower limits the integral value would assume the opposite sign. In our present case,
${\int }_0^{3}f\left(x\right).dx={\int }_3^{0}f\left(x\right).dxi.e.,{\int }_0^{3}f\left(x\right).dx=-{\int }_0^{3}f\left(x\right).dx\left(usingproperty\right)i.e.,2.{\int }_0^{3}f\left(x\right).dx=0i.e.,f\left(x\right)=c{|}_0^3=0$
Hence option (d) is correct.