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Let `I=int_(kpi)^((k+1)pi)(|sin2x|)/(|sinx|+|cosx|)dx,(kepsilonN)` and `J=int_(0)^((pi)/(4))(dx)/(sinx+cosx)` which of the following holds good?A. `I=2int_(0)^((pi)/(2))(sin2xdx)/(sinx+cosx)`B. `I=4-4J`C. `I=4-2J`D. `I=2-2J` |
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Answer» Correct Answer - A::B We have `I=int_(kpi)^((k+1)pi)(|sin2x|)/(|sinx|+|cosx|)dx` put `x=kpi+t` `impliesdx=dt` `I=int_(0)^(pi)(|sin2x|dx)/(|sinx|+|cosx|)=2int_(0)^((pi)/(2))(sin2xdx)/(sinx+cosx)` `=2int_(0)^((pi)/(2))((sinx+cosx)^(2)-1)/((sinx+cosx))dx` `2int_(0)^((pi)/(2))(sinx+cosx)dx-2int_(0)^((pi)/(2))(dx)/(sinx+cosx)` `=4-4int_(0)^((pi)/(4))(dx)/(sinx+cosx)=4-4J` |
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