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

Solution:

Given ∫ ln (x) dx = ∫ ln (x) . 1 dx

We use integration by parts to solve it i.e., ∫ u.v dx

= u ∫ v dx - ∫ [du/dx × ∫ v dx] dx

And to select the u and v we use the rule LIATE

L - logarithmic function

I - Inverse trigonometric

A - Algebraic function

T - Trigonometric function

E - Exponential function

∫ ln (x) .(1) dx

Here u = ln(x), v =1

= ln (x) ∫(1) dx - ∫[d(log x)/dx . ∫ 1. dx] dx

= ln (x) . x - ∫ (1/x). x .dx

= x ln(x) - x + C

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

Summary:

Integral of ln(x) dx using c for the constant of integration is x ln(x) - x + C