Question

The value of $\left\{ \frac{x}{2} \right\}$dx is (where {x} denotes the fractional part of x}-

• A. $\frac{- 2n\pi + 1}{\pi^{2}}$
• B. $\frac{n}{\pi}$
• C. $\frac{(n + 1)}{\pi}$
• D. $\frac{2n\pi - 1}{\pi^{2}}$

Answer 1

Answer: (A)

$\frac{- 2n\pi + 1}{\pi^{2}}$

Answer 2

I = $\int_{2n}^{2n + 1/2}{(\sin x)\left\{ \frac{x}{2} \right\} dx}$

= 2n $\int _ { 0 } ^ { 2 } ( \sin \pi x )$ $\frac{x}{2}$dx +$\int_{0}^{1/2}{(\sin\pi x)}\frac{x}{2}dx$

= $\frac{- 2n\pi + 1}{\pi^{2}}$