Question

Finding an Indefinite Integral dx/x(lnx^2)^3

Answer 1

Step-by-step explanation:

Firstly, we'll try to simplify the integrand. By hint 1, we see that:

$\ln(x^2) = 2\ln(x)$

Simplifying the integrand gives us:

$\frac{1}{8}\left(\frac{1}{x(\ln(x))^3}\right)$

Next, by hint 2, we observe that:

$\frac{d}{dx}\left(\ln(x)\right) = \frac{1}{x}$

So this tells us to make the substitution: $u = \ln(x)$

Doing so gives us:

$\int \frac{dx}{x(ln(x^2))^3} = \int \frac{du}{8u^3}$, which should be trivial.