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Evaluate: int((1-xsinx)dx)/(x(1-x^3e^(3c...

Evaluate: `int((1-xsinx)dx)/(x(1-x^3e^(3cosx)))`

Text Solution

Verified by Experts

Here, `I=int((1-xsinx)dx)/(x(1-(x e^(cosx))^(3)))`
Put `xe^(cosx)=t " so that " (xe^(cosx)(-sinx)+e^(cosx))dx=dt`
` :. I=int(dt)/(t(1-t^(3)))=int(dt)/(t^(4)((1)/(t^(3))-1))`
Let `(1)/(t^(3))-1=y " or " dy=(-3)/(t^(4))dt`
` :. I=(1)/(3)int (dy)/(y)=-(1)/(3)log|y|+C= -(1)/(3)log|(1)/(t^(3))-1|+C`
`= -(1)/(3)log_(e)|(1)/(x^(3)e^(3cosx))-1|+C`
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