Question

${\int }_0^a\left[f\right(a+x)+f(a-x\left)\right]d{x}_{=}$

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

Answer 1

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

${\int }_0^a\left[f\right(a+x)+f(a-x\left)\right]dx$......1
${\int }_0^{a}f(a+x)dx+{\int }_0^{a}f(a-x)dx$
${\int }_0^{a}f(a-x{)}^{dx}={\int }_0^{a}f(x)dx$
${\int }_0^{a}f(a+x)dx$

Let $t=a+x$
${\int }_0^{a}f(a+x)={\int }_a^{2a}f\left(t\right)\cdot dt={\int }_a^{2a}f\left(x\right)\cdot dx$ ---(2)
2 in 1
${\int }_0^{a}f\left(x\right)dx+{\int }_a^{2a}f\left(x\right)dx={\int }_0^{2a}f\left(x\right)dx$