Postgraduate advanced mathematics

Integral limit: x=0, t = π/4; x=π/4，t=0

Original formula = ∫ (0, π/4)(π/4-t)dt/[Costco(π/4-t)]

=∫(0，π/4)π/4dt/[Costco(π/4-t)]-∫(0，π/4)TDT/[Costco(π/4-t)]

Because different variables do not affect the final integral value, so:

∫(0，π/4)TDT/[Costco(π/4-t)]=∫(0，π/4) xdx/[cosxcos(π/4-x)]

Then: ∫ (0, π/4) xdx/[cosxcos (π/4-x)] =1/2 ∫ (0, π/4)π/4dt/[Costco(π/4-t)]

And ∫(0, π/4)π/4dt/[Costco(π/4-t)]

=∫(0，π/4)π/4dt/[Costcoπ/4 cost+sinπ/4 Sint]

=√2π/4∫(0，π/4) dt/[cost(cost+sint)]

=√2π/4∫(0，π/4) d(tant)/( 1+tant)

=√2π/4ln( 1+tant)|(0，π/4)

=√2πln2/4

Then: ∫ (0, π/4) xdx/[Cosxcos (π/4-x)] = √ 2π LN2/8.