We need to find the integral of cosec 2x
dx.
Int[ cosec 2x dx]
=> Int[ 1/
sin 2x dx]
=> Int[ (sin 2x)/ (sin 2x)^2
dx]
=> Int[ (sin 2x)/(1 - (cos 2x)^2)
dx]
=> Int[ (sin 2x)/(1 - cos 2x)(1 + cos 2x)
dx]
=> 1/2(Int [ sin 2x / (1 - cos 2x) + sin 2x/(1 + cos 2x)
dx]
=> 1/4[ ln| 1 - cos 2x | - ln|( 1 + cos
2x)]
=> 1/4[ - ln(|1 + cos 2x | / | 1 - cos 2x|)
]
=> 1/4[ - ln((1 + cos 2x)^2 / ((1 - cos 2x)^2)
]
=> 1/4[ - ln((1 + cos 2x)^2 / (sin
2x)^2]
=> 1/2[ - ln|(1/(sin 2x) + (cos 2x) / (sin
2x)|]
=> 1/2[ - ln|(cosec 2x) + (cot
2x)|]
=> -1/2( ln | cosec 2x + cot 2x| +
C
The required integral is -1/2(ln | cosec 2x + cot
2x| + C
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