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Question

If α0dx1cosαcosx=Asinα+B (α0) the values of A and B are


A

A=π2,B=0

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B

A=π4,B=π4sinα

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C

A=π6,B=πsinα

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D

A=π,B=πsinα

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Solution

The correct options are
A

A=π2,B=0


B

A=π4,B=π4sinα


I=α0dx1cosαcosx=α0dxcos2x2+sin2x2cosα(cos2x2sin2x2)=α0dx(1cosα)cos2x2+(1+cosα)sin2x2=α0sec2x2dx2sin2α2+2cos2α2tan2x2=12α0sec2α2sec2x2dxtan2α2+tan2x2
Let tanx2=tI=tanα/20sec2α2dtt2+tan2α2=sec2α2cotα2[tan1(ttanα2)]tanα20=2sinα.π4=π2sinαNow Asinα+B=π2sinαA=π2,B=0also A=π4,B=π4sinα
Hence (A), (B) are correct answer.


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