π0log(sin2x)dx=0?

1 Answer
Apr 12, 2018

I=ln(4)π

Explanation:

We want to solve

I=π0ln(sin2(x))dx

By the properties of logarithms

I=2π0ln(sin(x))dx

Use the trig identity

sin(x)=2sin(x2)cos(x2)

Thus

I=2π0ln(2sin(x2)cos(x2))dx

=2π0ln(2)dx+2π0ln(sin(x2))dx+2π0ln(cos(x2))dx

=ln(4)π+2π0ln(sin(x2))dx+2π0ln(cos(x2))dx

Make a substitution u=x2du=12dx
x=0u=0 and x=πu=π2

I=ln(4)π+4π20ln(sin(u))du+4π20ln(cos(u))du

Make a substitution u=π2sdu=ds
u=0s=π2 and u=π2s=π

I=ln(4)π+4π20ln(sin(u))du4ππ2ln(sin(s))ds

I=ln(4)π+4π20ln(sin(u))du4ππ2ln(sin(s))ds

Make a substitution w=sdw=ds
s=π2w=π2 and s=πw=π

I=ln(4)π+4π20ln(sin(u))du+4ππ2ln(sin(w))dw

I=ln(4)π+2I

I=ln(4)π

A very neat result