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Question

The value of the integral 10dxx2+2xcosα+1, α(0,π2) is equal to

A
sinα
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B
αsinα
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C
α2sinα
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D
α2sinα
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Solution

The correct option is C α2sinα
I=10dx(x+cosα)2+(1cos2α)
=10dx(x+cosα)2+sin2α
=1sinα[tan1x+cosαsinα]10
=1sinα[tan1(1+cosαsinα)tan1(cosαsinα)]
=1sinα[tan1(cotα2)tan1(cotα)]
=1sinα[tan1(tan(π2α2))tan1(tan(π2α))]
=1sinα [(π2α2)(π2α)]
=α2sinα

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