Let f ( x ) = x sin x ( 1 + x cos x . l n x + sin x ) d x and f ( 2 ) = 2 4 . Find the value of cos f.

Question

Let $f (x) = \int x^{\sin x} (1 + x \cos x . l n x + \sin x) d x$ and $f (\frac{π}{2}) = \frac{π^{2}}{4}$ . Find the value of $\cos (f (π))$ .

Smart Achievers · Accepted Answer

(2)

$f (x) = \int x^{\sin x} (1 + x \cos x . l n x + \sin x) d x$

$f (x) = x^{\sin x} = e^{\sin x . l n x}, then$

$f (x) = \int (f (x) + x f' (x)) d x = x \cdot x^{\sin x} + c$

$f (\frac{π}{2}) = \frac{π}{2} \cdot (\frac{π}{2}) + c \Rightarrow c = 0$

$f (x) = x (x^{\sin x}), f (π) = π .$

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