question_answer

Let a non-negative continuous function\[f(x)\]is such that the area of the region bounded by the curve\[y=f(x),\]\[x-\]axis and the lines\[x=\frac{\pi }{4}\]ands\[x=\beta >\frac{\pi }{4}\]is\[\left( \beta \sin \beta +\frac{\pi }{4}\cos \beta +\sqrt{2}\beta  \right),\]then the value of \[f\left( \frac{\pi }{2} \right)\]is

A)  \[\left( \frac{\pi }{4}+\sqrt{2}-1 \right)\]

B)  \[\left( \frac{\pi }{4}-\sqrt{2}+1 \right)\]

C)  \[\left( 1-\frac{\pi }{4}-\sqrt{2} \right)\]

D)  \[\left( 1-\frac{\pi }{4}+\sqrt{2} \right)\]