Let I=∫3π/4π/4ϕ1+sinϕdϕ=∫3π/4π/4(π−ϕ)1+sin(π−ϕ)dϕ=π∫3π/4π/411+sinϕdϕ−∫3π/4π/4ϕ1+sinϕdϕ
⇒2I=π∫3π/4π/411+sinϕdϕ⇒I=π2∫3π/4π/411+sinϕdϕ=π2∫3π/4π/411+cos(π2−ϕ)dϕ=π4∫3π/4π/4sec2(π4−ϕ2)dϕ=π4×−2[tan(π4−ϕ2)]3π/4π/4=−π2[tan(−π8)−tan(π8)]=π(√2−1)
Therefore
(√2+1)198π∫3π/4π/4ϕ1+sinϕdϕ=(√2+1)198π×π(√2−1)=198