Please do this one. Evaluate the integral [x(1-x)^-1*dx]Thank you  

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sciencesolve | Teacher | (Level 3) Educator Emeritus

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You need to evaluate the following indefinite integral such that:

`int x*(1-x)^(-1)dx = int x/(1-x) dx = -int -x/(1-x)dx`

You need to add and subtract 1 such that:

`-int -x/(1-x)dx = -int (1-x-1)/(1-x)dx`

You need to split the integral using the property of linearity, such that:

`-int (1-x-1)/(1-x)dx = -int (1-x)/(1-x)dx - int (-1)/(1-x)dx`

Reducing duplicate factors yields:

`-int (1-x-1)/(1-x)dx = - int dx + int 1/(1 - x) dx`

You need to come up with the following substitution such that:

`1 - x = t => -dx = dt => dx = -dt`

`-int (1-x-1)/(1-x)dx = - int dx - int (dt)/t`

`-int (1-x-1)/(1-x)dx = - x - ln|t| + c`

Substituting back `1 - x`  for `t `  yields:

`int x*(1-x)^(-1)dx = -x - ln|1 - x| + c`

Hence, evaluating the given indefinite integral yields `int x*(1-x)^(-1)dx = -x - ln|1 - x| + c.`

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