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Question

Assertion :Let F(x) be an indefinite integral of sin3/2xcos5/2x.
Statement 1: F is periodic function with period 2π. Reason: Statement 2: The period of sinnx and cosnx is 2π.

A
Both Statement 1 and Statement 2 are correct and Statement 2 is the correct explanation for Statement 1
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B
Both Statement 1 and Statement 2 are correct but Statement 2 is not the correct explanation for Statement 1
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C
Statement 1 is correct but Statement 2 is incorrect
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D
Both Statement 1 and Statement 2 are incorrect
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Solution

The correct option is C Statement 1 is correct but Statement 2 is incorrect
Let I=1sin3xcos2xdx
Multiply numerator and denominator by sec4x
I=sec4xtan32xdx=(1+tan2x)sec2xtan32xdx
Put t=tanxdt=sec2xdx
I=t2+1t32dt
Put u=tdu=12tdt
I=2u4+1u2du=2(u2+1u2)du
=2u332u=2(tan2x3)2tanx
=sin2xsin4x3sin3xcos5x
Now for sinnx,cosnx
If n is even period is π
And if n is odd period is 2π

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