Everyday Math 每日數學 之 Integration 積分 #20190218 +solution

#20190218

Solution:
(a)
$\int\frac{1}{\sin x + \cos x}dx$
$=\frac{\sqrt2}{2}\int\frac{1}{\frac{\sqrt2}{2}\sin x + \frac{\sqrt2}{2}\cos x}dx$
$=\frac{\sqrt2}{2}\int\frac{1}{\sin \left(x+\frac{\pi}{4}\right)}dx$
$=\frac{\sqrt2}{2}\int\csc\left(x+\frac{\pi}{4}\right)dx$
$=\frac{\sqrt2}{2}\int\frac{\csc\left(x+\frac{\pi}{4}\right)(\csc\left(x+\frac{\pi}{4}\right)+\cot\left(x+\frac{\pi}{4}\right))}{\csc\left(x+\frac{\pi}{4}\right)+\cot\left(x+\frac{\pi}{4}\right)}dx$
$=-\frac{\sqrt2}{2}\ln \left|\csc\left(x+\frac{\pi}{4}\right)+\cot\left(x+\frac{\pi}{4}\right)\right| +C$

(b)
$\int\frac{1}{\sin x + \sqrt3\cos x}dx$
$=\frac{1}{2}\int\frac{1}{\frac{1}{2}\sin x + \frac{\sqrt3}{2}\cos x}dx$
$=\frac{1}{2}\int\frac{1}{\sin \left(x+\frac{\pi}{3}\right)}dx$
$=\frac{1}{2}\int\csc\left(x+\frac{\pi}{3}\right)dx$
$=\frac{1}{2}\int\frac{\csc\left(x+\frac{\pi}{3}\right)(\csc\left(x+\frac{\pi}{3}\right)+\cot\left(x+\frac{\pi}{3}\right))}{\csc\left(x+\frac{\pi}{3}\right)+\cot\left(x+\frac{\pi}{3}\right)}dx$
$=-\frac{1}{2}\ln \left|\csc\left(x+\frac{\pi}{3}\right)+\cot\left(x+\frac{\pi}{3}\right)\right| +C$