Question

By using properties of definite integrals, evaluate the integrals
Show that
${\int }_0^{a}f\left(x\right)g\left(x\right)dx=2{\int }_0^{a}f\left(x\right)dx,$ $\text{if f and g are defined}$ $asf\left(x\right)=f(a-x)\text{and}g\left(x\right)+g(a-x)=4$

Answer 1

Let I $={\int }_0^{a}f\left(x\right)g\left(x\right)dx\Rightarrow I={\int }_0^{a}f(a-x)g(a-x)dx\Rightarrow I={\int }_0^{a}f\left(x\right)\{4-g(x\left)\right\}dx[\because f(x)=f(a-x\left)\text{and}g\right(x)+g(a-x)=4(given\left)\right]$
$\text{On adding Eqs, (i) and (ii), we get}$
$2I{\int }_0^{a}4f\left(x\right)dx\Rightarrow I=2{\int }_0^{a}f\left(x\right)dx.$