Question

Find the indefinite integral. (Remember to use absolute values where appropriate. Use C for the constant of integration.) dx x(ln x2)5

Answer 1

I'm assuming the integral is

`\displaystyle \int (dx)/(x (\ln(x^2))^5)`

We have

`\ln(x^2) = 2 \ln|x| \implies (\ln(x^2))^5 = 32 (\ln|x|)^5`

Then substituting
`y=\ln|x|` and
`dy=\frac{dx}x`, the integral transforms and reduces to

`\displaystyle \int (dx)/(x(\ln(x^2))^5) = \frac1{32} \int (dy)/(y^5) \\\\ ~~~~~~~~ = \frac1{32} \left(-\frac1{4y^4}\right) + C \\\\ ~~~~~~~~ = -\frac1x + C`

which we can rewrite as

`128 (\ln|x|)^4 = 8\cdot2^4(\ln|x|)^4 = 8 (2\ln|x|)^4 = 8 (\ln(x^2))^4`

and so

`\displaystyle \int (dx)/(x (\ln(x^2))^5) = \boxed{-\frac1{8(\ln(x^2))^4} + C}`