形式一

 $\begin{aligned} \int\csc\ x\ dx&=\int\frac1{\sin x}dx\\ &\xlongequal{二倍角公式}\int\frac1{2\sin\frac{x}2\cos\frac{x}2}dx\\ &\xlongequal{u=\frac{x}2}\int\frac1{\sin u\cos u} du\\ &\xlongequal{\sin u=\tan u\cos u}\int\frac1{\cos^2u\tan u}du\\ &\xlongequal{d(\tan u)=\sec^2u=\frac1{\cos^2u}}\int\frac1{\tan u}d\tan u\\ &=\ln|\tan u|+C\\ &=\ln|\tan\frac{x}2|+C \end{aligned}$

形式二

裂项参考：https://blog.iyatt.com/?p=20837

 $\begin{aligned} \int\csc\ x\ dx&=\int\frac1{\sin x}dx\\ &=\int\frac{\sin x}{\sin^2x}dx\\ &\xlongequal{d\cos x=-\sin x}-\int\frac1{\sin^2x}d\cos x\\ &\xlongequal{\sin^2x+\cos^2x=1}\int\frac1{\cos^2x-1}d\cos x\\ &=\int\frac1{(\cos x+1)(\cos x-1)}d\cos x\\ &\xlongequal{裂项}\frac12\int(\frac1{\cos x-1}-\frac1{\cos x+1})d\cos x\\ &=\frac12(\ln|\cos x - 1|)-\ln(\cos x+1)+C\\ &=\frac12\ln|\frac{\cos x-1}{\cos x+1}|+C \end{aligned}$

形式三

 $\begin{aligned} \int\csc x dx&=\int\csc x\frac{\csc x-\cot x}{\csc x-\cot x}dx\\ &=\int\frac{\csc^2x-\csc x\cot x}{\csc x-\cot x}dx\\ &\xlongequal[du=-\csc x\cot x+\csc^2x]{u=\csc x-\cot x}\int\frac1{u}du\\ &=\ln|u|+C\\ &=\ln|\csc x-\cot x|+C <----这是一种结果 \\ &\xlongequal[\csc x-\cot x=\frac1{\csc x+\cot x}]{\csc^2x-\cot^2x=1}\ln|(\csc x+\cot x)^{-1}|+C\\ &=-\ln|\csc x+\cot x|+C <----这是另外一种结果 \end{aligned}$

csc x 余割的不定积分计算