19.9k views
5 votes
Find the indefinite integral. (Remember to use absolute values where appropriate. Use C for the constant of integration.) dx x(ln x2)5

1 Answer

2 votes

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}

User Ilya Bibik
by
6.9k points
Welcome to QAmmunity.org, where you can ask questions and receive answers from other members of our community.