Integral of 1/(a sin x + b cos x): write the denominator as R sin(x + alpha)

1. A.
   $\sqrt{2}\log\left|\tan\!\left(\frac{x}{2}+\frac{\pi}{4}\right)\right|+c$
2. B.
   $\frac{1}{\sqrt{2}}\log\left|\tan\!\left(\frac{x}{2}+\frac{\pi}{8}\right)\right|+c$
3. C.
   $\frac{1}{\sqrt{2}}\log\left|\frac{\tan(x/2)-(\sqrt{2}+1)}{\tan(x/2)+(\sqrt{2}-1)}\right|+c$
4. D.
   $-\frac{1}{\sqrt{2}}\log\left|\frac{\tan(x/2)-(\sqrt{2}+1)}{\tan(x/2)+(\sqrt{2}-1)}\right|+c$

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1. A.
   $\dfrac{1}{2}\log\left|\csc\left(x - \dfrac{\pi}{3}\right) + \cot\left(x - \dfrac{\pi}{3}\right)\right| + c$
2. B.
   $\dfrac{1}{2}\log\left|\csc\left(x - \dfrac{\pi}{3}\right) - \cot\left(x - \dfrac{\pi}{3}\right)\right| + c$
3. C.
   $\dfrac{1}{2}\log\left|\csc\left(x + \dfrac{\pi}{3}\right) - \cot\left(x - \dfrac{\pi}{3}\right)\right| + c$
4. D.
   $\dfrac{1}{2}\log\left|\csc\left(x + \dfrac{\pi}{3}\right) - \cot\left(x + \dfrac{\pi}{3}\right)\right| + c$

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