MCQ
$\int_0^{\pi /2} {\frac{{\cos x}}{{1 + \cos x + \sin x}}} \,dx = $
- A$\frac{\pi }{4} + \frac{1}{2}\log 2$
- B$\frac{\pi }{4} + \log 2$
- ✓$\frac{\pi }{4} - \frac{1}{2}\log 2$
- D$\frac{\pi }{4} - \log 2$
$ = \int_0^{\pi /2} {\frac{{{{\cos }^2}(x/2) - {{\sin }^2}(x/2)}}{{2{{\cos }^2}(x/2) + 2\sin (x/2)\cos (x/2)}}} dx$
$ = \frac{1}{2}\int_0^{\pi /2} {\frac{{1 - {{\tan }^2}(x/2)}}{{1 + \tan (x/2)}}} dx = \frac{1}{2}\int_0^{\pi /2} {\left[ {1 - \tan \left( {\frac{x}{2}} \right)} \right]} dx$
$\frac{\pi }{4} + \log \frac{1}{{\sqrt 2 }} = \frac{\pi }{4} - \frac{1}{2}\log 2$
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If the derivative $f^{\prime}$ of $f$ satisfies the equation $f ^{\prime}( x )=\frac{ f ( x )}{ b ^2+ x ^2}$ for all $x \in R$, then which of the following statements is/are TRUE?
$(A)$ If $b>0$, then $f$ is an increasing function
$(B)$ If $b<0$, then $f$ is a decreasing function
$(C)$ $(x)(-x)=1$ for all $x \in R$
$(D)$ $(x)-f(-x)=0$ for all $x \in R$