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

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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