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Q.

The value of   sin2xsin4x+cos4xdx can be equal to

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a

tan1(cos2x)+C

b

cot1(cot2x)+C

c

cot1(tan2x)+C

d

tan1(tan2x)+C

answer is A, B, C, D.

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Detailed Solution

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I=sin2xsin4x+cos4xdx=2tanxsec2xtan4x+1dx    Put  tan2x=t    2tanxsec2xdx=dt=dtt2+1=tan1(t)+C=π2cot1(tan2x)+C

=cot1(tan2x)+C1=cot1(1cot2x)+C1=cot1(cot2x)+C1

Also  cos2x=1tan2x1+tan2x(1cos2x1+cos2x)=tan2x, using these values in given integral

I=sin2x(cos2xsin2x)2+2sin2xcos2xdx=2sin2x(cos2x)2+1dx

Put  cos2x=t2sin2xdx=dt

I=dtt2+1=tan1t+C2=tan1(cos2x)+C2

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