Question

The value of $\int_{–2n}^{2n + \frac{1}{2}}{(\sin\pi x)\left\{ \frac{x}{2} \right\} dx}$ 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 = $\int_{–2n}^{2n + \frac{1}{2}}{(\sin\pi x)\left\{ \frac{x}{2} \right\} dx}$

=2n

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