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Question

Obtain the reduction formula for In=cosecn x dx, n being a positive integer, n2 and hence deduce the value of cosec5 x dx.

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Solution

In=cscnx=cscn2xcsc2x
By integration by parts , we get In=cscn2x(cotx)(cotx)(n2)cscn3x(cscx)cotxdx=cscn2xcotx(n2)cscn2xcot2x
In=cscn2xcotx(n2)cscn2x(csc2x1)=cscn2xcotxn1+n2n1cscn2x
Therefore the value of csc5x=csc3xcotx4+34csc3x
=csc3xcotx4+34(cscxcotx2+12cscx)

=csc3xcotx4+3cscxcotx4+3ln(cscx+cotx)8

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