Definite Integration Question 250
Question: The derivative of $ F(x)=\int _{x^{2}}^{x^{3}}{\frac{1}{\log t}dt} $ , $ (x>0) $ is
Options:
A) $ \frac{1}{3\log x}-\frac{1}{2\log x} $
B) $ \frac{1}{3\log x} $
C) $ \frac{3x^{2}}{3\log x} $
D) $ {{(\log x)}^{-1}}.x(x-1) $
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Answer:
Correct Answer: D
Solution:
We know that $ \frac{d}{dx}( \int_a^{b}{f(t)dt} )=\frac{db}{dx}f(b)-\frac{da}{dx}f(a) $
a and b are functions of x.
$ \therefore F(x)=\int _{x^{2}}^{x^{3}}{\frac{1}{\log t}dt} $
therefore $ F’(x)=\frac{d}{dx}(x^{3})\frac{1}{\log x^{3}}-\frac{d}{dx}(x^{2})\frac{1}{\log x^{2}} $
$ =\frac{3x^{2}}{3\log x}-\frac{2x}{2\log x}=x(x-1){{(\log x)}^{-1}} $ .
BETA
