If I = ∫ sin ⁡ x 3 sin ⁡ x + cos ⁡ x + 2 d x and J = ∫ cos ⁡ x 3 sin ⁡ x + cos ⁡ x + 2 d x , then 3 J - I is equal to (where C is the constant of integration)

Question

If I = ∫ sin ⁡ x 3 sin ⁡ x + cos ⁡ x + 2 d x and J = ∫ cos ⁡ x 3 sin ⁡ x + cos ⁡ x + 2 d x , then 3 J - I is equal to (where C is the constant of integration), x + ln ⁡ 3 sin ⁡ x + cos ⁡ x + 2 + c, x - ln ⁡ 3 sin ⁡ x + cos ⁡ x + 2 + c, ln ⁡ 3 sin ⁡ x + cos ⁡ x + 2 + c, 2 x + ln ⁡ 3 sin ⁡ x + cos ⁡ x + 2 + c

MARKS App · Accepted Answer

3 J - I = ∫ 3 c o s x - s i n x 3 s i n x + c o s x + 2 d x Let 3 s i n x + c o s x + 2 = t ⇒ 3 cos ⁡ x - sin ⁡ x d x = d t So, 3 J - I = ∫ d t t = ln ⁡ t + C Hence, 3 J - I = ln ⁡ 3 sin ⁡ x + cos ⁡ x + 2 + c

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