Silution;
Substitute;
u=8x
So that ;
dxdu=8
dx=81du
Substitute the values back to the equation;
81∫08π1+cos(u)du
Apply the half angle identity;
cos(u)+1=2cos2(2u )
By substitution;
81∫08π2cos2(2u)
Simplify as;
81∫08π2cos(2u)du
Take;
v=2u ;dudv=21 ;du=2dv
By substitution;
42∫04πcos(v)dv
Integrate;
[42sin(v)]04π
But v=2u ,also u=8x ,hence v=4x
Replace back the value of v;
[42sin(4x)]04π
42[sin(4π)−sin(0)] =0
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