Question

Evaluate the integral

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

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

Answer 1

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

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

Using the property ${\int }_a^{b}f\left(x\right)dx={\int }_a^{b}f(a+b-x)dx$ in the second part:-

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

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

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

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

Hence, answer is option-(B).