Question

${\int }_0^{\frac{\pi }{4}}\sin (x-[x\left]\right)d(x-[x\left]\right)$ is equal to :

• A. $\frac{1}{2}$
• B. $1-\frac{1}{\sqrt{2}}$
• C. $1$
• D. None of these

Answer 1

The correct option is B $1-\frac{1}{\sqrt{2}}$

Given : ${\int }_0^{\frac{\pi }{4}}\sin (x-[x\left]\right)d(x-[x\left]\right)$

Let :

$t=x-\left[x\right]x\to 0t\to 0dt=d(x-[x\left]\right)x\to \frac{\pi }{4}t\to \frac{\pi }{4}{\int }_0^{\frac{\pi }{4}}\sin tdt=-{\left[\cos t\right]}_0^{\frac{\pi }{4}}=1-\frac{1}{\sqrt{2}}$

Hence the correct answer is $1-\frac{1}{\sqrt{2}}$