Let Ι = ∫ c o s 3 x 1 + s i n 2 x d x , then Ι is equal to (where c is the constant of integration)

Question

Let Ι = ∫ c o s 3 x 1 + s i n 2 x d x , then Ι is equal to (where c is the constant of integration), 2 t a n - 1 x + sin ⁡ x + c, 2 t a n - 1 sin ⁡ x - sin ⁡ x + c, 2 t a n - 1 x - x + c, 2 t a n - 1 sin ⁡ x + sin ⁡ x + c

MARKS App · Accepted Answer

Let, sin ⁡ x = t ⇒ cos ⁡ x d x = d t Ι = ∫ c o s 2 x 1 + t 2 d t = ∫ 1 - s i n 2 x 1 + t 2 d t = ∫ 1 - t 2 1 + t 2 d t or Ι = ∫ 2 - 1 + t 2 1 + t 2 d t = ∫ 2 d t 1 + t 2 - ∫ d t = 2 t a n - 1 t - t + c = 2 t a n - 1 sin ⁡ x - sin ⁡ x + c

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