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Question

If f(x)=(cotx2tanx2)dx, where f(π2)=0, then which of the following statements is (are) CORRECT ?

(Note : sgn(y) denotes the signum function of y.)

A
π0f(x) dx=2πln2
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B
π0f(x) dx=πln2
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C
sgn(f(2π3))=1
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D
sgn(f(2π3))=1
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Solution

The correct option is C sgn(f(2π3))=1
f(x)=(cotx2tanx2)dx

f(x)=⎜ ⎜cosx2sinx2sinx2cosx2⎟ ⎟dx

f(x)=⎜ ⎜cos2x2sin2x2sinx2cosx2⎟ ⎟dx

f(x)=2⎜ ⎜cos x2sinx2cosx2⎟ ⎟dx
=2cotx dx
=2ln|sinx|+C

At x=π2,f(π2)=0
C=0
f(x)=2ln|sinx|=lnsin2x

π0f(x) dx
=π02ln(sinx)dx
=22π/20ln(sinx)dx
=4(π2ln2)=2πln2

Now,f(2π3)=ln34=ve
sgn(f(2π3))=1

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