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Let us take
z = a + b cosx
∴ dz = - b sinx dx
or, sinx dx = - (dz)/b
∴ ∫ (a + b cosx)ⁿ sinx dx
= ∫ zⁿ {- (dz)/b}
= - 1/b ∫ zⁿ dz
= - 1/b (zⁿ⁺¹)/(n + 1) + c, where c is integral constant and n ± (- 1)
= - 1/{b (n + 1)} (a + b cosx)ⁿ⁺¹ + c
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2
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