The value of ∫0π/2 sin⁡2θsin⁡θdθ is

I=∫0π/2 sin⁡2θsin⁡θdθ=∫0π/2 sin⁡(π−2θ)sin⁡(π/2−θ)dθ=∫0π/2 sin⁡2θcos⁡θdθ2I=∫0π/2 sin⁡2θ(sin⁡θ+cos⁡θ)dθIn 2I, put sin⁡θ−cos⁡θ=t, so that t2=1−sin⁡2θ⇒1−t2=sin⁡2θ. Also dt=(cos⁡θ+sin⁡θ)dθ∴ 2I=∫−11 1−t2dt∣=2t21−t2+12sin−1⁡t01=2π4=π2⇒ I=π4.