MCQ
$\int_0^1 {{{\tan }^{ - 1}}x\,dx = } $
- ✓$\frac{\pi }{4} - \frac{1}{2}\log 2$
- B$\pi - \frac{1}{2}\log 2$
- C$\frac{\pi }{4} - \log 2$
- D$\pi - \log 2$
$\Rightarrow dx = {\sec ^2}\theta \,\,d\theta $
Also as $x = 0,\theta = 0$ and $x = 1,\theta = \frac{\pi }{4}$
Therefore, $\int_0^1 {{{\tan }^{ - 1}}x\,dx = \int_0^{\pi /4} {\theta {{\sec }^2}\theta \,d\theta } } $
$ = \frac{\pi }{4}$$ - \log \sqrt 2 = \frac{\pi }{4} - \frac{1}{2}\log 2$.
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$S_n(x)=\sum_{k=1}^n \cot ^{-1}\left(\frac{1+k(k+1) x^2}{x}\right)$
where for any $x \in R , \cot ^{-1} x \in(0, \pi)$ and $\tan ^{-1}(x) \in\left(-\frac{\pi}{2}, \frac{\pi}{2}\right)$. Then which of the following
statements is (are) $TRUE$?
$(A)$ $S _{10}( x )=\frac{\pi}{2}-\tan ^{-1}\left(\frac{1+11 x ^2}{10 x }\right)$, for all $x >0$
$(B)$ $\lim _{n \rightarrow \infty} \cot \left(S_n(x)\right)=x$, for all $x>0$
$(C)$ The equation $S_3(x)=\frac{\pi}{4}$ has a root in $(0, \infty)$
$(D)$ $\tan \left( S _{ n }( x )\right) \leq \frac{1}{2}$, for all $n \geq 1$ and $x >0$