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Question Number 82244 by M±th+et£s last updated on 19/Feb/20

f i n d t h e f u n c t i o n o f f w h e n t h i s

f u n c t i o n c o n t i n u e a t i n t e r v a l [- \infty, 0]

\int_{- x^{2}}^{0} f (t) d t = \frac{d}{d x} [x (1 - s i n (π x)]

Commented by mr W last updated on 19/Feb/20

\int_{- x^{2}}^{0} f (t) d t = \frac{d}{d x} [x (1 - s i n (π x)]

\int_{- x^{2}}^{0} f (t) d t = 1 - s i n (π x) - π x \cos (π x)

2 x f (- x^{2}) = - 2 π \cos (π x) + π^{2} x \sin (π x)

f (- x^{2}) = - \frac{π}{x} \cos (π x) + \frac{π^{2}}{2} \sin (π x)

t = - x^{2} \Rightarrow x = \pm \sqrt{- t}

f (t) = \mp \frac{π}{\sqrt{- t}} \cos (π \sqrt{- t}) \pm \frac{π^{2}}{2} \sin (π \sqrt{- t})

\Rightarrow f (x) = \mp \frac{π}{\sqrt{- x}} \cos (π \sqrt{- x}) \pm \frac{π^{2}}{2} \sin (π \sqrt{- x})

Commented by M±th+et£s last updated on 19/Feb/20

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Answered by mind is power last updated on 19/Feb/20

\Rightarrow 2 x f (- x^{2}) = \frac{d^{2}}{{d x}^{2}} [x - x s i n (π x)]

\Rightarrow f (- x^{2}) = \frac{1}{2 x} \frac{d}{d x} [- s i n (π x) - π x c o s (π x)]

\Rightarrow f (- x^{2}) = \frac{1}{2 x} . [- 2 π c o s (π x) + π^{2} x s i n (π x)]

f (- x^{2}) = - \frac{π c o s (π x)}{x} + \frac{π^{2}}{2} s i n (π x)

- x^{2} = y \Rightarrow x = \underline{+} \sqrt{- y}

w e g o t 2 s o l u t i o n

f (y) = - \frac{π c o s (π \sqrt{- y})}{\sqrt{- y}} + \frac{π^{2} s i n (π \sqrt{- y})}{2} y \neq 0

f (0) = 0

a n d - f

Commented by M±th+et£s last updated on 19/Feb/20

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