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Let $f:(0,\infty)\rightarrow R$ and $f(x)=\int_0^xf(t) dt$ if $f(x^2)=x^2(1+x)$ then f(4) equals

$(a)\;5/4\qquad(b)\;7\qquad(c)\;4\qquad(d)\;2$

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$F(x)=\int_0^x f(t)dt$
$\Rightarrow f'(x)=f(x)-f(0)$
Also $F(x^2)=x^2(1+x)$
$\Rightarrow F'(x^2)2x=2x+3x^2$
$\therefore F'(4)=f(4)$
$f(0)=0$
$F'(4)\times 4=4+12$
$F'(4)=4$
$\Rightarrow f(4)=4$
Hence (c) is the correct answer.
answered Jan 2, 2014 by sreemathi.v
 

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