If ∫sec2x−2010sin2010xdx=P(x)sin2010x+C,then value of P(π3) is
A
0
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B
1√3
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C
√3
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D
Noneofthese
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Solution
The correct option is C√3 ∫sec2x−2010sin2010xdx=∫sec2x(sinx)−2010−2010∫1(sinx)2010dx=I1−I2 Applying by parts on I1, we get I1=tanx(sinx)2010+2010∫tanxcosx(sinx)2011dx=tanx(sinx)2010+2010∫dx(sinx)2010⇒I=I1−I2=tanx(sinx)2010=P(x)(sinx)2010P(π3)=tanπ3=√3