Evaluate the integral. (use c for the constant of integration.) (ln(x))² dx

Solution:

Given function is (ln(x))² dx

Let Integral I =∫(ln(x))² dx ---------->(1)

Let us use integration by parts.

∫u. dv = uv - ∫v. du

∫u. dv = (ln(x))² dx

Let u = (ln(x))² and dv = dx

⇒du = 2 . ln x .1/x dx and v = x

Thus  (1) becomes

= (ln(x))². x - ∫x. 2 . ln x .1/x dx 

= x (ln(x))²-  ∫2 . ln x . dx 

= x (ln(x))²-  2 . (xln x -x)

= x (ln(x))²-  2x.ln x + 2x + C

Thus ∫(ln(x))² dx = x (ln(x))²-  2x.ln x + 2x + C

Evaluate the integral. (use c for the constant of integration.) (ln(x))² dx

Summary:

Evaluating the integral (using c for the constant of integration) (ln(x))2 dx, we get x (ln(x))²-  2x.ln x + 2x + C