MCQ
The value of $\int_0^1 {\frac{{dx}}{{{e^x} + {e^{ - x}}}}} $ is
- A${\tan ^{ - 1}}\left( {\frac{{1 - e}}{{1 + e}}} \right)$
- ✓${\tan ^{ - 1}}\left( {\frac{{e - 1}}{{e + 1}}} \right)$
- C$\frac{\pi }{4}$
- D${\tan ^{ - 1}}e + \frac{\pi }{4}$
Now put ${e^x} = t \Rightarrow {e^x}dx = dt$
Also as $x = 0$ to $1$, $t = 1$ to $e$, then reduced form is
$\int_1^e {\frac{{dt}}{{1 + {t^2}}} = [{{\tan }^{ - 1}}t]_1^e} = {\tan ^{ - 1}}\left( {\frac{{e - 1}}{{e + 1}}} \right)$,
$\left[ \because {{\tan }^{-1}}x-{{\tan }^{-1}}y={{\tan }^{-1}}\left( \frac{x-y}{1+xy} \right) \right]$.
Generate a complete, print-ready paper with questions like this in minutes — across 16+ boards, with answer keys.
Which of the following statements is/are $TRUE$ ?
$(A)$ No $a$ satisfies the above equation
$(B)$ An integer $a$ satisfies the above equation
$(C)$ An irrational number $a$ satisfies the above equation
$(D)$ More than one $a$ satisfy the above equation