Question

If $f\left(x\right)$ is a periodic function with period $T$, then ${\int }_{a+2T}^{b+2T}f\left(x\right)dx$ is equal to

• A. ${\int }_{a+T}^{b+T}f\left(x\right)dx$
• B. ${\int }_{a+T}^{b}f\left(x\right)dx$
• C. ${\int }_a^{b+T}f\left(x\right)dx$
• D. ${\int }_a^{b}f\left(x\right)dx$

Answer 1

Answer: (A), (D)

The correct options are
A ${\int }_{a+T}^{b+T}f\left(x\right)dx$
D ${\int }_a^{b}f\left(x\right)dx$

${\int }_{a+2T}^{b+2T}f\left(x\right)dx$
$={\int }_{a+T}^{b+T}f(y+T)dy$ $($ Putting $x=y+T$ so that $dx=dy)$

$={\int }_a^{b}f\left(z\right)dz$ $($ Again, putting $z=y+T$ so that $dz=dy)$

$={\int }_a^{b}f\left(x\right)dx$

Hence, (A) and (D) are correct