The value of the integral ∫0π/23cosθcosθ+sinθ5dθ is equal to

I=∫0π/23cosθ(cosθ+sinθ)5dθ=∫0π/23sinθ(sinθ+cosθ)5dθ2I=∫0π/23(sinθ+cosθ)4dθ=∫0π/23sec2θ(1+Tanθ)4dθ Put, Tanθ=t2⇒sec2θdθ⇒2tdt=∫0∞6t(1+t)4dt=6∫0∞1(1+t)3-1(1+t)4dt⇒I=3(1+t)-33-(1+t)-220∞   ⇒    I=313(0-1)-12(0-1)=12