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Question Number 84861 by Jakir Sarif Mondal last updated on 16/Mar/20
Iff(x)isanevenfunction,then∫π0f(cosx)dx=2∫π/20f(cosx)dx
Commented by mr W last updated on 17/Mar/20
f(x)iseven,⇒f(−x)=f(x)∫0πf(cosx)dx=∫0π2f(cosx)dx+∫π2πf(cosx)dx=I1+I2letx=t+π2⇒t=x−π2dx=dtπ2⩽x⩽π⇒0⩽t⩽π2cosx=cos(t+π2)=−costf(cosx)=f(−cost)=f(cost)I2=∫π2πf(cosx)dx=∫0π2f(cost)dt=I1⇒∫0πf(cosx)dx=2I1=2∫0π2f(cosx)dx
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