Question

If ${\int }_0^{2a}f\left(x\right)dx=2{\int }_0^{a}f\left(x\right)dx$ then which of the following statements is true?

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

Answer 1

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

${\int }_0^{2a}f\left(x\right)dx={\int }_0^{a}f\left(x\right)dx+{\int }_a^{2a}f\left(x\right)dx={\int }_0^{a}f\left(x\right)dx+{\int }_0^{a}f(2a-x)dx$

Therefore given equation converts to ${\int }_0^{a}f\left(x\right)dx+{\int }_0^{a}f(2a-x)dx=2{\int }_0^{a}f\left(x\right)dx$

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

Therefore option $A$ is correct