Question
$$\int_0^{\frac{\pi }{4}} {\sin \,x\,d\left( {x - \left[ x \right]} \right)} $$ is equal to :
A.
$$\frac{1}{2}$$
B.
$$1 - \frac{1}{{\sqrt 2 }}$$
C.
1
D.
none of these
Answer :
$$1 - \frac{1}{{\sqrt 2 }}$$
Solution :
$$\eqalign{
& {\text{Let }}x - \left[ x \right] = z.{\text{ In }}0 \leqslant x \leqslant \frac{\pi }{4},\,\left[ x \right] = 0.\,\,{\text{So }}x = z. \cr
& \therefore I = \int_0^{\frac{\pi }{4}} {\sin \,z\,dz} \,\, = \left[ { - \cos \,z} \right]_0^{\frac{\pi }{4}}\,\, = 1 - \frac{1}{{\sqrt 2 }} \cr} $$