Question

If $\int\limits_0^{50 \pi} \frac{ x |\sin x |}{\{ x \}+\{- x \}} dx = k \pi( k \in N )$, 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)

$I=\int\limits_0^{50 \pi} \frac{x|\sin x|}{\{x\}+\{-x\}} d x=\int\limits_0^{50 \pi} \frac{x|\sin x|}{1} d x ;$

Using king & add, we get
$2 I=50 \pi \int\limits_0^{50 \pi}|\sin x| d x $
$I=25 \pi \times 50 \int_0^\pi|\sin x| d x=2500 \pi \Rightarrow k=2500$