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Let f(x) be a non negative continuous function such that area bounded by the curve y = f(x), the x-axis and the ordinates $x = \frac{π}{4}$ and $x = β > \frac{π}{4}$ is $β \sin β + \frac{π}{4} \cos β + \sqrt{2} β$ . Then $f (\frac{π}{2})$ =

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$\frac{π}{4} - \sqrt{2} + 1$

$\frac{π}{4} + \sqrt{2} - 1$

$1 - \frac{π}{4} + \sqrt{2}$

$1 + \frac{π}{4} + \sqrt{2}$

answer is C.

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Detailed Solution

$\int_{\frac{π}{4}}^{β} f (x) d x = β \sin β + \frac{π}{4} \cos β + \sqrt{2} β$

Ditterentiating w.r.t $β$ ,

$f (β) = \sin β + β \cos β - \frac{π}{4} \sin β + \sqrt{2}$

$f (\frac{π}{2}) = 1 - \frac{π}{4} + \sqrt{2}$

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