Evaluate the following integral: `int_0^pixsinxcos^4x\ dx`

Explanation

Given: `int_0^pi xsinxcos^4x dx`

`I=int_0^pixsinxcos^4x dx......(i)`

`I=int_0^pi (pi-x)sin(pi-x)cos^4(pi-x )dx`

`I=int_0^pi (pi-x)sinxcos^4xdx.....(ii)`

Now add equation (i) and (ii)

`2I=int_0^pi xsinxcos^4xdx + int_0^pi (pi-x) sinxcos^4x dx`

`2I=int_0^pi pisinxcos^4x dx`

Let `cosx=t implies -sinx dx=dt`and limits changes as `x = 0 implies t = 1, x=pi implies t = -1`

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