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Question

Evaluate: π/3π/3π+4x32cos(|x|+π3)dx

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Solution

aaf(x)dx=a0(f(x)+f(x))dx
π3π3π+4x32cos(|x|+π3)=π30π+4x32cos(|x|π3)+π4x32cos(|x|+π3)
I=2ππ30dx2(x+π3)
x+π3=z
I=2π2π3π3dz2cosZ
=2π2π3π3dz1+2sin2z2
2π2π2π3secz2sec2z2+2tan2z2=2π323sec2z2dz1+3tanz2
=4π3tan1(3tanz2)2π2π3
tan1(3tanπ3)tan1(3tanπ3)
tan1(3)tan1(1)
tan1θ=tan1(311+3)=tan112
4π3tan112.

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