Consider the integral A= displaystyle ∫ 01 (ex - 1/x)dx and B= displaystyle ∫ 01(x/ex - 1)dx . Then, which of the following is incorrect?

Question

Consider the integral A= displaystyle ∫ 01 (ex - 1/x)dx and B= displaystyle ∫ 01(x/ex - 1)dx . Then, which of the following is incorrect? (A) B < 1 (B) A>1 (C) B>A (D) A>(1/2)

Tardigrade Admin · Accepted Answer

As, ex-1>x∀ x∈ (0 , ∈ fty) ∴ (ex - 1/x)>1 Hence, A= displaystyle ∫ 01(ex - 1/x)dx> displaystyle ∫ 011dx ⇒ A>1 And, B= displaystyle ∫ 01 (x/ex - 1)dx < displaystyle ∫ 01 1dx ⇒ B < 1 ∴ B < 1 < A

Q. Consider the integral $A = \int_{0}^{1} \frac{e ^{x} - 1}{x} d x$ and $B = \int_{0}^{1} \frac{x}{e ^{x} - 1} d x$ . Then, which of the following is incorrect?

$B < 1$

$A > 1$

$B > A$

$A > \frac{1}{2}$

As, $e^{x} - 1 > x \forall x \in (0, \in f t y)$
$∴ \frac{e ^{x} - 1}{x} > 1$
Hence, $A = \int_{0}^{1} \frac{e ^{x} - 1}{x} d x > \int_{0}^{1} 1 d x$
$\Rightarrow A > 1$
And, $B = \int_{0}^{1} \frac{x}{e ^{x} - 1} d x < \int_{0}^{1} 1 d x$
$\Rightarrow B < 1$
$∴ B < 1 < A$

Q. Consider the integral A=∫01​xex−1​dx and B=∫01​ex−1x​dx . Then, which of the following is incorrect?

B<1

A>1

B>A

A>21​