How to integrate (ln x)²?

To find Integration of (ln x)², we will use integration by parts method.

Answer: The final integral of (ln x)² is x (ln x)² − 2x ln x + 2x + C.

Go through the explanation to understand better.

Explanation:

Given:

y = (ln x)²

∫(ln x)²dx = ∫(ln x) (ln x)dx

Let u = ln x, dv = ln x dx

So, 

du = 1 / x dx and v =x ln x − x

Using Integration by Parts,

∫(ln x)²dx = (ln x)(x ln x − x) − ∫(x lnx−x) / x dx

= x (ln x)² − x ln x − ∫(ln x − 1)dx

= x (ln x)² − x ln x − (x ln x− x − x) + C

= x (ln x)² − 2x ln x + 2x + C

Thus, the final integral of (ln x)² is x (ln x)² − 2x ln x + 2x + C