Mathematics - Definite Integration Question with Solution | TestHub

Let $I={\int }_0^{a}f\left(x\right)g\left(x\right)dx$

$\Rightarrow I={\int }_0^{a}f\left(a-x\right)g\left(a-x\right)dx$    (Using properties of definite integration)

$\Rightarrow I={\int }_0^{a}f\left(x\right)\left(4-g\left(x\right)\right)dx \left[\because f\left(x\right)=f\left(a-x\right) \& g\left(x\right)+g\left(a-x\right)=4\right]$

$\Rightarrow I=4{\int }_0^{a}f\left(x\right)dx-{\int }_0^{a}f\left(x\right)g\left(x\right)dx$

$\Rightarrow I=4{\int }_0^{a}f\left(x\right)dx-I$

$\Rightarrow 2I=4{\int }_0^{a}f\left(x\right)dx$

$\Rightarrow I=2{\int }_0^{a}f\left(x\right)dx$