First slide

Introduction to integration

Question

If $f (x)$ is the primitive of
$\frac{\sin \sqrt[3]{x} \log (1 + 3 x)}{{(\tan^{- 1} \sqrt{x})}^{2} (e^{\sqrt[3]{x}} - 1)} (x \neq 0) then lim_{x \to 0} f^{'} (x)$ is

Moderate

A

0
B

$3 / 5$
C

$\frac{5}{3}$
D

none of these

Solution

$lim_{x \to 0} f^{'} (x) = lim_{x \to 0} \frac{\sin \sqrt[3]{x} \log (1 + 3 x)}{{(\tan^{- 1} \sqrt{x})}^{2} (e^{\sqrt[3]{x}} - 1)}$ \
$\begin{array}{l} = lim_{x \to 0} 3 \frac{\sin \sqrt[3]{x}}{\sqrt[3]{x}} \times \frac{\log (1 + 3 x)}{3 x} \times {(\frac{\sqrt{x}}{\tan^{- 1} \sqrt{x}})}^{2} \frac{\sqrt[3]{x}}{e^{\sqrt[3]{x}} - 1} \\ = 3 (1) (1) (1)^{2} (1) = 3 \end{array}$

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