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Question

# If ∫α0dx1−cosαcosx=Asinα+B (α≠0) 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=∫α0dx1−cosαcosx=∫α0dxcos2x2+sin2x2−cosα(cos2x2−sin2x2)=∫α0dx(1−cosα)cos2x2+(1+cosα)sin2x2=∫α0sec2x2dx2sin2α2+2cos2α2tan2x2=12∫α0sec2α2sec2x2dxtan2α2+tan2x2 Let tanx2=tI=∫tanα/20sec2α2dtt2+tan2α2=sec2α2cotα2[tan−1(ttanα2)]tanα20=2sinα.π4=π2sinαNow Asinα+B=π2sinα⇒A=π2,B=0also A=π4,B=π4sinα Hence (A), (B) are correct answer.

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