Question

If $\underset{0}{\overset{50\pi }{\int }}\frac{x|\sin x|}{\{x\}+\{-x\}}dx=k\pi (k\in N)$, then the value of $k$ is
[Note: $\{y\}$ denotes fractional part of $y$.]

• A. 1250
• B. 2500
• C. 5000
• D. 5050

Answer 1

Answer: (B)

$I=\underset{0}{\overset{50\pi }{\int }}\frac{x|\sin x|}{\{x\}+\{-x\}}dx=\underset{0}{\overset{50\pi }{\int }}\frac{x|\sin x|}{1}dx;$

Using king & add, we get
$2I=50\pi \underset{0}{\overset{50\pi }{\int }}|\sin x|dx$
$I=25\pi \times 50{\int }_0^{\pi }|\sin x|dx=2500\pi \Rightarrow k=2500$