Question

Identify the correct representation of integral of even function on a symmetric interval. ${\int }_{-a}^{a}f\left(x\right)dx$

• A. ${\int }_0^{a}f\left(x\right)dx$
• B. $2{\int }_{-a}^{a}f\left(x\right)dx$
• C. $2{\int }_0^{a}f\left(x\right)dx$
• D. None of these

Answer 1

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

Given : ${\int }_{-a}^{a}f\left(x\right)dx$

Since $f\left(x\right)$ is even function $f\left(x\right)=f(-x)$

Then,

${\int }_{-a}^{0}f\left(x\right)dx={\int }_0^{a}f\left(x\right)dx{\int }_{-a}^{a}f\left(x\right)dx={\int }_{-a}^{0}f\left(x\right)dx+{\int }_0^{a}f\left(x\right)dx=2{\int }_0^{a}f\left(x\right)dx$