How to integrate the following? 1/{sin^2 (x) cos^2 (x)}

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Answer 1 · 2018-04-18T15:11:48

#int1/(sin^2xcos^2x)dx=-2cot2x+c#

#int1/(sin^2xcos^2x)dx#

= #int4/(4sin^2xcos^2x)dx#

= #4int1/(sin^2 2x)dx#

= #4intcsc^2 2xdx#

Let #u=2x#, then our integral becomes

#4intcsc^2u(du)/2#

= #2intcsc^2udu#

= #2(-cotu+c_1)#

= #-2cotu+2c_1#

= #-2cot2x+c#, where #c# is another constant so that #c=2c_1#