given-f-x-x-2-sin-2x-0-pi-4-f-x-dx-f-pi-2-

Question Number 77842 by jagoll last updated on 11/Jan/20

g i v e n

f (x) = x^{2} + \sin (2 x) + \int \underset{0}{\overset{\frac{π}{4}}{}} f (x) d x

f (\frac{π}{2}) = ?

Answered by mr W last updated on 11/Jan/20

l e t \int_{0}^{\frac{π}{4}} f (x) d x = a = c o n s t a n t

f (x) = x^{2} + \sin (2 x) + a

\int_{0}^{\frac{π}{4}} f (x) d x = {[\frac{x^{3}}{3} - \frac{\cos (2 x)}{2} + a x]}_{0}^{\frac{π}{4}} = a

\frac{π^{3}}{192} + \frac{1}{2} + \frac{a π}{4} = a

\Rightarrow a = \frac{π^{3} + 96}{48 (4 - π)}

f (x) = x^{2} + \sin (2 x) + \frac{π^{3} + 96}{48 (4 - π)}

f (\frac{π}{2}) = \frac{π^{2}}{4} + \frac{π^{3} + 96}{48 (4 - π)} = \frac{(48 - 11 π) π^{2} + 96}{48 (4 - π)}

Commented by jagoll last updated on 11/Jan/20

t h a n k s y o u s i r

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