Consider the integrals I = ∫ sin ⁡ x 3 cos ⁡ x + 4 sin ⁡ x d x and J = ∫ cos ⁡ x 3 cos ⁡ x + 4 sin ⁡ x d x . Then, the integral 4 J - 3 I is equal to (where, c is the constant of integration)

Step-by-step solution

4 J - 3 I = ∫ 4 cos ⁡ x - 3 sin ⁡ x 3 cos ⁡ x + 4 sin ⁡ x d x Let, 3 cos ⁡ x + 4 sin ⁡ x = t ⇒ 4 cos ⁡ x - 3 sin ⁡ x d x = d t ⇒ 4 J - 3 I = ∫ d t t = ln ⁡ t + c ⇒ 4 J - 3 I = ln ⁡ 3 cos ⁡ x + 4 sin ⁡ x + c