Graduado en Matemáticas Avanzadas

Supongamos π/4-x=t, entonces dx=-dt.

Límite integral: x=0, t =π/4; x=π/4, t=0

Fórmula original = ∫ (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)]

Debido a que diferentes variables no afectan el valor integral final, entonces:

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

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

Y ∫ (0,π/4)π/4dt/[Costco(π/4-t)]

=∫(0,π/4)π/4dt/[Costcoπ/4 costo+senπ/4 Sint]

=√2π/4∫(0,π/4) dt/[costo(costo+sint)]

=√2π/4∫(0,π/ 4) d(tant)/(1+tant)

=√2π/4ln(1+tant)|(0,π/4)

=√2πln2/4

Entonces: ∫ (0, π/4)xdx/[cos xcos(π/4-x)]=√2πLN2/8.