Question

integration of cot square x/(csc square x+cscx)dx
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Answer 1

x + cos x + c

Step-by-step explanation:

Integrate cot²x/(cosec²x+cosec x)dx

Multiply Nr and Dr by the conjugate i.e. (cosec²x - cosec x),

= integrate cot²x*(cosec²x-cosec x) / (cosec²x+cosec x)*(cosec²-cosec x) dx

= integrate cot²x*(cosec²x - cosec x) / (cosec⁴x - cosec²x) dx

= integrate cot²x*(cosec²x - cosec x) / cosec²x*(cosec²x - 1) dx [As cosec²x-1 = cot²x]

=integrate cot²x*(cosec²x-cosec x) / cosec²x*(cot²x)dx

=integrate (cosec²x-cosec x) / cosec²x dx

=integrate cosec²x / cosec²x dx - integrate cosec x /cosec²x dx

= x - integrate (1/cosec x) dx

= x - integrate sin x dx [Integration sin x = - cos x+c]

= x - (- cos x) + c

= x + cos x + c