The integral I = ∫ sin 3 ⁡ θ cos ⁡ θ 1 + sin 2 ⁡ θ 2 d θ simplifies to (where, c is the constant of integration)

Question

The integral I = ∫ sin 3 ⁡ θ cos ⁡ θ 1 + sin 2 ⁡ θ 2 d θ simplifies to (where, c is the constant of integration), 1 2 ln ⁡ sin ⁡ θ + 1 1 + sin 2 ⁡ θ + c, 1 2 ln ⁡ 1 + sin 2 ⁡ θ + 1 1 + sin 2 ⁡ θ + c, ln ⁡ sin ⁡ θ + 1 1 + sin 2 ⁡ θ + c, ln ⁡ sin 2 ⁡ θ + 1 + 1 sin 2 ⁡ θ + 2 + c

MARKS App · Accepted Answer

Let 1 + sin 2 ⁡ θ = t ⇒ 2 sin ⁡ θ cos ⁡ θ d θ = d t ∴ I = 1 2 ∫ t - 1 t 2 d t = 1 2 ∫ 1 t - 1 t 2 d t = 1 2 ln ⁡ t + 1 t + c = 1 2 ln ⁡ 1 + sin 2 ⁡ θ + 1 1 + sin 2 ⁡ θ + c

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