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Home  >>  JEEMAIN and AIPMT  >>  Mathematics  >>  Class12  >>  Integral Calculus
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$f(x^2) =x^3 (2+x^2),$$ f(x^2)= \int \limits_0^{x^3} f(t) dt,$ find $f(8)=?$

\[\begin {array} {1 1} (a)\;8 \\ (b)\;9 \\ (c)\;10 \\ (d)\;12 \end {array}\]
Can you answer this question?
 
 

1 Answer

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$f(x^2) =x^3 (2+x^2)$
$f(x^2)=\int \limits_0^{x^3} f(t).dt$
$x^3(2+x^2)= \int \limits_0^{x^3} f(t).dt$
By using levniJ theorem:
$3x^2.2 + 6x^5 =f(x^3).3x^2$
$3x^2(2 +3x^2) =f(x^3). 3x^2$
$2(1+x^2) =f(x^3)$
$f(x^3)=2 (1+x^2)$
Put $x=2$
$f (8)=2(1+2^2)$
$\qquad= 10$
answered Dec 16, 2013 by meena.p
 

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