Question

If $f(a+x)=f\left(x\right)$ , then $\forall$ $a>0,nϵN$ the value of ${\int }_0^{na}f\left(x\right)dx$ equals ?

• A. $(n-1){\int }_0^{a}f\left(x\right)dx$
• B. $(1-n){\int }_0^{a}f\left(x\right)dx$
• C. $n{\int }_0^{a}f\left(x\right)dx$
• D. None of the above

Answer 1

The correct option is A $(n-1){\int }_0^{a}f\left(x\right)dx$

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

Substitute $x=a+t\Rightarrow dx=dt$

$I={\int }_0^{(n-1)a}f(a+t)dt=(n-1){\int }_0^{a}f(a+t)dt=(n-1){\int }_0^{a}f(a+x)dx=(n-1){\int }_0^{a}f\left(x\right)dx$