The integral ∫ sin 2 x cos 2 x sin 5 x + cos 3 x sin 2 x + sin 3 x cos 2 x + cos 5 x 2 d x , is equal to (where C is the constant of integration).

Step-by-step solution

∫ sin 2 x cos 2 x sin 5 x + cos 3 x sin 2 x + sin 3 x cos 2 x + cos 5 x 2 d x = ∫ sin 2 x cos 2 x sin 3 x sin 2 x + cos 2 x + cos 3 x   cos 2 x + sin 2 x 2 d x   = ∫ sin 2 x cos 2 x sin 3 x + cos 3 x 2 d x = ∫ sin 2 x cos 2 x cos 3 x 2 tan 3 x + 1 2 d x = ∫ tan 2 x sec 2 x tan 3 x + 1 2 d x Put, tan 3 x + 1 = t ⇒ d t = 3 tan 2 x sec 2 x d x = ∫ d t 3 t 2 = - 1 3 t + C , where C is the constant of integration. =     - 1 3 tan 3 x + 1 + C