Let $f:(0,\infty) \rightarrow R$ and $F(x) = \int_{0}^{x}{f(x)dt.}$ If $F\left( x^{2} \right) = x^{2}(1 + x),$ then $f(4)$ equals

Question

Dr. Abubacker · Accepted Answer

(c)
Given, $F\left( x^{2} \right) = x^{2}(1 + x)$
$\Rightarrow \ \int_{0}^{x^{2}}{f(t)dt = x^{2}(1 + x)}$
On differentiating both sides w.r.t., we get
$2xf\left( x^{2} \right) = 2x + 3x^{2}$
$\Rightarrow f\left( x^{2} \right) = 1 + \frac{3x}{2}$
$\Rightarrow f\left( 2^{2} \right) = 1 + \frac{3}{2}(2) = 4\  \Rightarrow \ f(4) = 4$

Q. Let $f:(0,\infty) \rightarrow R$ and $F(x) = \int_{0}^{x}{f(x)dt.}$ If $F\left( x^{2} \right) = x^{2}(1 + x),$ then $f(4)$ equals

Explanation

Reviewed by Dr. Abubacker

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