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Question

Which of the following is/are equal to sin5x dx
(where C is integration constant)

A
15cosxsin4x+415cosx45cosx+C
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B
cos5x5+2cos3x3cosx+C
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C
15cosxsin4x415cosxsin2x815cosx+C
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D
cos5x5+2cos3x3cosx+C
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Solution

The correct option is C 15cosxsin4x415cosxsin2x815cosx+C
I=sin5xdx=sin4xsinxdx=(1cos2x)2sinxdx
Let, t=cosx
dt=sinxdx=(1t2)2dt=(1+t42t2)dt=[t+t552t33]+c=cosxcos5x5+2cos3x3+c=cos5x5+2cos3x3cosx+c
Now,
In=sinnxdx=cosxsinn1xn+n1nIn2
I5=cosxsin4x5+45I3=cosxsin4x5+45[cosxsin2x3+23I1]=cosxsin4x5415cosxsin2x+815sinx dx=15cosxsin4x415cosxsin2x815cosx+c

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