Question

If$f\left(x\right)=f(a+x)$ and ${\int }_0^{a}f\left(x\right)dx=p$, then ${\int }_a^{na}f\left(x\right)dx$ is equal to

• A. $np$
• B. $(n-1)p$
• C. $(n+1)p$
• D. none of these

Answer 1

Answer: (B)

$(n-1)p$

Since, $f\left(x\right)=f(a+x)$, therefore $a$ is the period of $f\left(x\right)$

Again, ${\int }_{ma}^{na}f\left(x\right)dx=(n-m){\int }_0^{a}f\left(x\right)dx$

Here, ${\int }_a^{na}f\left(x\right)dx=(n-1){\int }_0^{a}f\left(x\right)dx$

$=(n-1)p$

Hence, answer is option-(B).