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We need to determine the integral of (cos x)^7 * sin x.
Int [(cos x)^7 * sin x dx]
let cos x = u => - du = sin x dx
=> Int [ -u^7 du]
=> -u^8 / 8 + C
substitute u = cos x
=> - (cos x)^8 / 8 + C
The required integral is - (cos x)^8 / 8 + C
So, we'll have to calculate the indefinite integral of the function (cos x)^7*sin x.
Int (cos x)^7*sin x dx
We'll solve the indefinite integral using substitution technique.
We'll put cos x = t =>-sin x dx = dt
We'll raise to 7th power cos x:
(cos x)^7 = t^7
We'll re-write the integral:
-Int t^7 dt = -t^8/8 + c
We'll substitute t by cos x:
Int (cos x)^7*sin x dx = -(cos x)^8/8 + C
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