Question

If $f\left(x\right)=f(a-x)$, then ${\int }_0^{a}f\left(x\right)dx$ is equal to

• A. ${\int }_0^{a}f\left(x\right)dx$
• B. $\frac{a^2}{2}{\int }_0^{a}f\left(x\right)dx$
• C. $\frac{a}{2}{\int }_0^{a}f\left(x\right)dx$
• D. $-\frac{a}{2}{\int }_0^{a}f\left(x\right)dx$

Answer 1

The correct option is C $\frac{a}{2}{\int }_0^{a}f\left(x\right)dx$

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