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  4. If x φ(x)=∫ limits5x(3 t2-2 φ prime(t)) d t, x>-2, and φ(0)=4, then φ(2) is

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Q. If $x \phi(x)=\int\limits_{5}^{x}\left(3 t^{2}-2 \phi^{\prime}(t)\right) d t, x>-2$, and $\phi(0)=4$, then $\phi(2)$ is ___

JEE MainJEE Main 2021Integrals

Solution:

$x \phi(x)=\int\limits_{5}^{x} 3 t^{2}-2 \phi^{\prime}(t) d t$
$x \phi(x)=x^{3}-125-2[\phi(x)-\phi(5)]$
$x \phi(x)=x^{3}-125-2 \phi(x)-2 \phi(5)$
$\phi(0)=4 $
$\Rightarrow \phi(5)=-\frac{133}{2}$
$\phi(x)=\frac{x^{3}+8}{x+2}$
$\phi(2)=4$