MCQ
$\int {\frac{{\cos x}}{{\cos (x - a)}}dx - } \int {\frac{{\sin x}}{{\sin (x - a)}}dx = } $
- A$2x\, \cos\, a + c$
- B$\sin\, a \,\log |\tan(x - a)| + c$
- C$\log |\cot (x - a)| c$
- ✓None of these
$\mathrm{I}_{1}=\int \frac{\cos \mathrm{x}}{\cos (\mathrm{x}-\mathrm{a})} \mathrm{dx} \Rightarrow \int \frac{\cos (\mathrm{x}-\mathrm{a}+\mathrm{a})}{\cos (\mathrm{x}-\mathrm{a})} \mathrm{dx}$
$\Rightarrow \int \frac{\cos (x-a) \cos a-\sin (x-a) \sin a}{\cos (x-a)} d x$
$\Rightarrow \cos a \int d x-\sin a \int \tan (x-a) d x$
$\mathrm{I}_{1} \Rightarrow \mathrm{x} \cos \mathrm{a}+\sin \mathrm{a} \log |\cos (\mathrm{x}-\mathrm{a})|+\mathrm{C}_{1} $ .......$(2)$
and $ \mathrm{I}_{2}=\int \frac{\sin \mathrm{x}}{\sin (\mathrm{x}-\mathrm{a})} \mathrm{d} \mathrm{x}$
$=\int \frac{\sin (x-a+a)}{\sin (x-a)} d x \Rightarrow \int \frac{\sin (x-a) \cos a+\cos (x-a) \sin a}{\sin (x-a)}$
$\Rightarrow \cos a \int d x+\sin a \int \cot (x-a) d x$
$I_{2}=x \cos a+\sin a \log |\sin (x-a)|+C_{2}$ ......$(3)$
by eq $(1)$
$\mathrm{I}=\mathrm{x} \cos \mathrm{a}+\sin \mathrm{a} \log \cos (\mathrm{x}-\mathrm{a})-\mathrm{x} \cos \mathrm{a}-$
$\sin a \log \sin (x-a)+C$
$=\sin a \log \left|\frac{\cos (x-a)}{\sin (x-a)}\right|+C$
$=\sin a \log |\cot (x-a)|+C$
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$\left[\frac{x}{\sqrt{x^{2}-y^{2}}}+e^{\frac{y}{x}}\right] x \frac{d y}{d x}=x+\left[\frac{x}{\sqrt{x^{2}-y^{2}}}+e^{\frac{y}{x}}\right] y$
pass through the points $(1,0)$ and $(2 \alpha, \alpha), \alpha>0$.
Then $\alpha$ is equal to