Question

If $nϵN$, then the value of ${\int }_{-n}^n(-1{)}^{\left\{x\right\}}dx$ (where $\{.\}$ denotes fractional part), is equal to

• A. 0
• B. 1
• C. $2n$
• D. $n$

Answer 1

The correct option is A 0

Let $g\left(x\right)=(-1{)}^{\left\{x\right\}}$
$g(-x)=(-1{)}^{\{-x\}}=(-1{)}^{-\left\{x\right\}}=-(-1{)}^{\left\{x\right\}}=-g\left(x\right)$
$I={\int }_{-n}^n(-1{)}^{\left\{x\right\}}dx=0$