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Question: The derivative of \(F(x) = \int_{x^{2}}^{x^{3}}{\frac{1}{\log t}dt}\), \((x > 0)\) is...

The derivative of F(x)=∫x2x31log⁡tdtF(x) = \int_{x^{2}}^{x^{3}}{\frac{1}{\log t}dt}, (x>0)(x > 0) is

A

13log⁡x−12log⁡x\frac{1}{3\log x} - \frac{1}{2\log x}

B

13log⁡x\frac{1}{3\log x}

C

3x23log⁡x\frac{3x^{2}}{3\log x}

D

(log⁡x)−1.x(x−1)(\log x)^{- 1}.x(x - 1)

Answer

(log⁡x)−1.x(x−1)(\log x)^{- 1}.x(x - 1)

Explanation

Solution

We know that

ddx(∫abf(t)dt)=dbdxf(b)−dadxf(a)\frac{d}{dx}\left( \int_{a}^{b}{f(t)dt} \right) = \frac{db}{dx}f(b) - \frac{da}{dx}f(a)

a and b are functions of x.

∴F(x)=∫x2x31log⁡tdt\therefore F(x) = \int_{x^{2}}^{x^{3}}{\frac{1}{\log t}dt}

⇒ F′(x)=ddx(x3)1log⁡x3−ddx(x2)1log⁡x2F'(x) = \frac{d}{dx}(x^{3})\frac{1}{\log x^{3}} - \frac{d}{dx}(x^{2})\frac{1}{\log x^{2}}

=3x23log⁡x−2x2log⁡x=x(x−1)(log⁡x)−1= \frac{3x^{2}}{3\log x} - \frac{2x}{2\log x} = x(x - 1)(\log x)^{- 1}.