The value of the integral ∫ 0 π / 2 3 cos ⁡ θ cos ⁡ θ + sin ⁡ θ…

The value of the integral ∫0π/23cos⁡θcos⁡θ+sin⁡θ5dθ equals

Solution

I=∫0π23cos⁡θsin⁡θ+cos⁡θ5.dθ …(i)
I=∫0π23sin⁡θcos⁡θ+sin⁡θ5.dθ …(ii) using ∫abfx.dx=∫abfa+b-x.dx
(i) + (ii)
⇒2I=∫0π23sin⁡θ+cos⁡θsin⁡θ+cos⁡θ5.dθ
⇒2I=3∫0π2dθsin⁡θ+cos⁡θ4
⇒2I3=∫0π2sec2⁡θ.dθ1+tan⁡θ4
Put tan⁡θ=t2
sec2⁡θ.dθ=2t.dt
when
θ→0,t→0
θ→π2,t→∞
⇒2I3=∫0∞2tdt1+t4
⇒I3=∫0∞t+1-11+t4.dt
I3=∫0∞11+t3-11+t4.dt
I3=-121+t2+131+t30∞
I3=0+0--12+13
I3=16⇒I=12

Asked in: JEE Advanced 2019 (Paper 2)

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