The value of ∫ sin ⁡ 101 x . sin 99 ⁡ x d x is sin ⁡ 100 x   sin 100 ⁡ x k + 5 , then k 19 is

Step-by-step solution

I = ∫ sin ⁡ 100 x ⋅ cos ⁡ x + cos ⁡ 100 x ⋅ sin ⁡ x sin 99 ⁡ x ⋅ d x I = ∫ sin ⁡ 100 x ⋅ cos ⁡ x . sin 99 ⁡ x d x + ∫ cos ⁡ 100 x ⋅ sin 100 ⁡ x ⋅ d x I I I I = sin ⁡ 100 x sin 100 ⁡ x 100 - 100 100 ∫ cos ⁡ 100 x sin 100 ⁡ x d x + ∫ cos ⁡ 100 x . sin 100 ⁡ x ⋅ d x I = sin ⁡ 100 x ⋅ sin 100 ⁡ x 100 + C