integration help!!!!!!!!!!!!!!!!!!!!

Question

 

Intergate 8 cot^3(x)?

Answer 1

integral 8 cot³(x)dx = 8  ∫cot²(x)dx

integral 8 cot³(x) =dx 8  ∫cot²(x) * cot(x) dx

Recall : Trigonometry formulas csc²(x) - cot²(x) = 1 Then csc²(x) - 1 = cot²(x)

Substitute cot²(x) = csc²(x) - 1   in the integral equation

integral 8 cot³(x)dx = 8  ∫[(csc²(x) -1] * cotx dx

Recall : Distributive property (a - b) * c =(a*c) - (b*c)

integral 8 cot³(x) dx= 8 ∫[(csc²(x) * cot(x) - 1 * cot(x)]dx

Recall : Commutative property of multiplication is a*b = b*a

integral 8 cot³(x) = 8 ∫cot(x) * csc²(x) - cot(x)] dx 

By the integral formula ∫[f(x) + g(x)] = ∫f(x) + ∫g(x)                     ∫

integral 8 cot³(x) = 8 [∫[cot(x) * csc²(x) dx -  ∫cot(x) dx]

put cot(x) = t Then dt = -csc²(x) dx There fore -dt = csc²(x) dx

Substitute cot(x) = t , csc²(x) dx = -dt in the ∫[cot(x) * csc²(x) dx

integral 8 cot³(x) = 8 [ ∫[t(-dt) -  ∫cot(x) dx ]

integral 8 cot³(x) = 8 [-∫ [tdt] -  ∫cot(x) dx ]

integral 8 cot³(x) = 8 [-t² / 2] -  ∫cot(x) dx ]

Substitute t = cotx in the above equation

integral 8 cot³(x) = 8 [-cot²(x) / 2 -  ∫cot(x) dx ]

Substitute int ∫cot(x) dx = logsin(x) in the above equation

integral 8 cot³(x) = 8 [-cot²(x) / 2 - logsinx ]

Take out common term negitive one

integral 8 cot²(x) = -8 [(cot²(x)) / 2 + logsin(x)]