Question

Consider the integral $A=\displaystyle \int _{0}^{1} \frac{e^{x} - 1}{x}dx$ and $B=\displaystyle \int _{0}^{1}\frac{x}{e^{x} - 1}dx$ . Then, which of the following is incorrect?

• A. $B < 1$
• B. $A>1$
• C. $B>A$
• D. $A>\frac{1}{2}$

Answer 1

Answer: (C)

As, $e^{x}-1>x\forall x\in \left(0 , \in fty\right)$
$\therefore \frac{e^{x} - 1}{x}>1$
Hence, $A=\displaystyle \int _{0}^{1}\frac{e^{x} - 1}{x}dx>\displaystyle \int _{0}^{1}1dx$
$\Rightarrow A>1$
And, $B=\displaystyle \int _{0}^{1} \frac{x}{e^{x} - 1}dx < \displaystyle \int _{0}^{1} 1dx$
$\Rightarrow B < 1$
$\therefore B < 1 < A$