∫x{f(x2)g′′(x2)−f′′(x2)g(x2)}dx

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Question

\int x {f (x^{2}) g^{''} (x^{2}) - f^{''} (x^{2}) g (x^{2})} d x

Options:

A . f(x2)g′(x2)−g(x2)f′(x2)+c

B . 12{f(x2)g(x2)f′(x2)}+c

C . 12{f(x2) g′(x2)−g(x2) f′(x2)}+c

D . None of these

Answer: Option C
:
C
Put

x^{2} = t \Rightarrow I = \frac{1}{2} \int {f (t) g^{''} (t) - g (t) f^{''} (t)} d t = \frac{1}{2} {f (t) g^{^{'}} (t) - \int f^{^{'}} (t) g^{^{'}} (t) - g (t) f^{^{'}} (t) + \int g^{^{'}} (t) f^{^{'}} (t) d t} + c ∴ I = \frac{1}{2} {f (t) g^{^{'}} (t) - g (t) f^{^{'}} (t)} + c = \frac{1}{2} {f (x^{2}) g^{^{'}} (x^{2}) - g (x^{2}) f^{^{'}} (x^{2})} + c

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