Question

If $L=\underset{x \rightarrow \frac{\pi}{2}}{\text{Lim}} \int\limits_{\frac{\pi}{2}}^x \frac{\cos ^2 t \sqrt{a^2-\cos ^2 t}}{b x+a \cot x-\frac{b \pi}{2}} d t$ exists finitely, then

• A. $L=0$ if $a \neq b$
• B. $L=1$ if $a=b, a<0$
• C. $L=-1$ if $a \neq b$
• D. $L=-1$ if $a=b, a>0$

Answer 1

Answer: (A), (B), (D)

$L=\underset{x \rightarrow \frac{\pi}{2}}{\text{Lim}} \int\limits_{\frac{\pi}{2}}^x \frac{\cos ^2 t \sqrt{a^2-\cos ^2 t}}{b x+a \cot x-\frac{b \pi}{2}} d t \quad\left(\frac{0}{0}\right.$ form $)$
$=\underset{x \rightarrow \frac{\pi}{2}}{\text{Lim}} \frac{\cos ^2 x \sqrt{a^2-\cos ^2 x}}{b-a \operatorname{cosec}^2 x}=0 \text { if } a \neq b$
when $a = b$
$L=\underset{x \rightarrow \frac{\pi}{2}}{\text{Lim}}\frac{\cos ^2 x \sqrt{a^2-\cos ^2 x}}{a\left(1-\operatorname{cosec}^2 x\right)}=\underset{x \rightarrow \frac{\pi}{2}}{\text{Lim}} \frac{\cos ^2 x \sqrt{a^2-\cos ^2 x}}{-a \cot ^2 x}$
$=\underset{x \rightarrow \frac{\pi}{2}}{\text{Lim}} \frac{\sin ^2 x \sqrt{a^2-\cos ^2 x}}{-a}=\frac{|a|}{-a}=\begin{cases}-1 ; & a >0 \\ 1 ; & a< 0\end{cases}$