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Question

If  $$\displaystyle \mathrm{F}(\mathrm{x})=\int_{x^{2}}^{x^{3}}\log tdt(\mathrm{x}>0)$$ , then $$\mathrm{F}^{1}(\mathrm{x})=$$


A
(9x24x)logx 
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B
(4x9x2)logx
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C
(9x2+4x)logx 
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D
(4x29x)logx
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Solution

The correct option is A $$(9\mathrm{x}^{2}-4\mathrm{x})\log x$$ 

$$F(x)=\int_{x^{2}}^{x^{3}}log\ t\ dt\ \ \ (x>0)$$ then
$$F^{'}(x)$$
$$F^{'}(x)=\left [ log\ x^{3}\cdot \dfrac{dx^{3}}{dx}-log\ x^{2}\dfrac{dx^{2}}{dx} \right ]$$
$$=[3\ log\ x(3x^{2})-2\ log\ x\ 2x]$$
$$=9(log\ x)x^{2}-h\ log\ x\ (x)$$
$$=(9x^{2}-4x)log\ x$$


Mathematics

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