First slide

Area of bounded Regions

Question

$If y = \int_{x^{2}}^{x^{3}} \frac{1}{\log t} d t (where x > 0), then find \frac{d y}{d x}$

Moderate

A

$x (x - 1) (\log x)^{- 1}$
B

$(x - 1) (\log x)^{- 1}$
C

$x (x - 1) (\log x)$
D

$x (\log x)^{- 1}$

Solution

$\begin{matrix} y = \int_{x^{2}}^{x^{3}} \frac{1}{\log t} d t \\ ∴ \frac{d y}{d x} = \frac{d}{d x} (x^{3}) \frac{1}{\log x^{3}} - \frac{d}{d x} (x^{2}) \frac{1}{\log x^{2}} \\ = \frac{3 x^{2}}{3 \log x} - \frac{2 x}{2 \log x} = x (x - 1) (\log x)^{- 1} \end{matrix}$

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