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### Integral of Ln x

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Date: 3/31/96 at 13:28:34
From: Anonymous
Subject: Calculus

Dr. Math,

Can you please tell me what the integral of the natural log of
x is (ln x)?
```

```
Date: 3/31/96 at 22:27:25
From: Doctor Sebastien
Subject: Re: Calculus

Hello,

Here's the question: Integral (ln x) dx

Let's substitute ln x by y:

x = e^y

dx/dy = e^y

So

Integral (ln x) dx = Integral (ln x) dx/dy dy = Integral (y e^y) dy

Using integration by parts,

Integral (y e^y) dy = uv - Integral (v du/dy) dy

Let u = y.

du/dy = 1

Integral dv = Integral (e^y) dy

v = e^y

Therefore,

Integral (y e^y) dy = y e^y - Integral (e^y * 1) dy

= y e^y - (e^y) + c,

where c is a constant. Therefore,

Integral (ln x) dx = y e^y - (e^y) + c

= (ln x)e^(ln x) - e^(ln x) + c

= x(ln x) - x + c

Sometimes you will do integrations of simple terms that don't
have a solution that take only one line.  You will then have to use
integration by parts.

-Doctor Sebastien,  The Math Forum

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Associated Topics:
High School Exponents

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