How solve integral 1/x(ln x+2)^4 ?

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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 1/(x(ln x+2)^4)dx`

You should come up with the following substitution, such that:

`ln x + 2 = t => 1/x dx = dt`

Replacing the variable, yields:

`int (dt)/t^4 = int t^(-4)dt`

`int t^(-4)dt = t^(-4+1)/(-4+1) + c`

Replacing back `ln x + 2` for t yields:

`int 1/(x(ln x+2)^4)dx = -1/(3(ln x + 2)^3) + c`

Hence, evaluating the indefinite integral, using substitution, yields `int 1/(x(ln x+2)^4)dx = -1/(3(ln x + 2)^3) + c.`

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