What is the integral of cos^5 x?  

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We have to find the integral of f(x) = (cos x)^5.

Int [f(x)] = Int [(cos x)^4 cos x dx]

=> Int [(1 – (sin x)^2)^2 cos x dx]

let t = sin x

=> dt/dx = cos x,

=> dt = cos x dx.

Int [(1 – (sin x)^2)^2 cos x dx]

=> Int [(1 – t^2)^2 dt]

=> Int [1 + t^4 – 2t^2 dt]

=> Int [1 dt] – Int [2t^2] + Int [t^4]

=> t – (2/3)*t^3 + t^5/5 +C

replace t with sin x

=> sin x – (2/3)*(sin x)^3 + (sin x) ^5 /5 + C

Therefore the required integral of cos^5 x is sin x – (2/3)*(sin x)^3 + (sin x)^5 /5 + C.

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