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int0^pisin^3x(1+2cosx)(1+cosx)^2dx...

`int_0^pisin^3x(1+2cosx)(1+cosx)^2dx`

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Given:`int_0^pisin^3x(1+2cosx)(1+cosx)^2dx`
`I=int_0^pisin^3x(1+2cosx)(1+cosx)^2dx`
`I=int_0^pisin^2x(1+2cosx)(1+cosx)^2 sinxdx`
`I=int_0^pi(1-cos^2x)(1+2cosx)(1+cosx)^2 sinxdx`
Let `cosx=t implies -sinxdx=dt implies sinxdx=-dt` and limits changes as when` x=0,implies t=1`, when `x=pi, implies t=-1`
`I=int_1^(-1) (1-t^2)(1+2t)(1+t)^2 (-dt)`
`I=-int_1^(-1) (1-t^2)(1+2t)(1+t)^2 dt`
`I=-int_1^(-1) (1+2t-t^2-2t^3)(1+2t+t^2)dt`
...
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