∫π0log(sin2x)dx=0?
1 Answer
Apr 12, 2018
Explanation:
We want to solve
I=∫π0ln(sin2(x))dx
By the properties of logarithms
I=2∫π0ln(sin(x))dx
Use the trig identity
sin(x)=2sin(x2)cos(x2)
Thus
Make a substitution
I=ln(4)π+4∫π20ln(sin(u))du+4∫π20ln(cos(u))du
Make a substitution
I=ln(4)π+4∫π20ln(sin(u))du−4∫−π−π2ln(−sin(s))ds
I=ln(4)π+4∫π20ln(sin(u))du−4∫−π−π2ln(sin(−s))ds
Make a substitution
I=ln(4)π+4∫π20ln(sin(u))du+4∫ππ2ln(sin(w))dw
I=ln(4)π+2I
I=−ln(4)π
A very neat result