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JEE Advanced 2010Paper 1

Exam: JEE Advanced
Year: 2010
Paper: Paper 1
Subject: Mathematics
Chapter: Limits

JEE Advanced 2010 · Paper 1 · Mathematics · Limits

The value of _ x arrow 0 (1)/(x^3) ∫_0^x (t (1+t))/(t^4+4) d t is

The value of

\lim _{x \rightarrow 0} \frac{1}{x^3} \int_0^x \frac{t \log (1+t)}{t^4+4} d t

is

Options

A0
B $\frac{1}{12}$
C $\frac{1}{24}$
D $\frac{1}{64}$

Correct answer

B. $\frac{1}{12}$

Step-by-step solution

\lim _{x \rightarrow 0} \frac{1}{x^3} \int_0^{x t} \frac{\log (1+t)}{4+t^4} d t

Using L' Hospital's rule,

\begin{aligned}& =\lim _{x \rightarrow 0} \frac{\frac{x \log (1+x)}{4+x^4}}{3 x^2}\\ = & \lim _{x \rightarrow 0} \frac{\log (1+x)}{3 x} \cdot \frac{1}{4+x^4}\\ = & \frac{1}{3} \cdot \frac{1}{4}=\frac{1}{12}\left[\text{ using, } \lim _{x \rightarrow 0} \frac{\log (1+x)}{x}=1\right] \end{aligned}

$

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