Proof of the integral of lnx and other examples involving the integration of natural logs.


In this hub I will show how to integrate lnx and other functions involving lnx. The best way to integrate lnx is to use integration by parts.

That is the ∫uv’ dx= uv - ∫u’vdx

If you haven’t used integration parts before then you will need to practice this before you attempt the integral of lnx.

Example 1

Work out ∫lnxdx

First rewrite this as ∫lnx.1dx

So u = lnx and v’ = 1. The reason you make u = lnx instead of u = 1 is that you should know the differential of lnx is 1/x. Therefore u’ = 1/x. Also if you integrate 1 you get x so v = x.

Next substitute u = lnx, u’ = 1/x, v = x and v’ = 1 into the above formula for integration by parts:

∫uv’ dx= uv - ∫u’vdx

∫lnx.1dx = lnx.x - ∫(1/x).xdx

Next tidy this up and (1/x) × x will cancel to give 1:

=xlnx - ∫1.dx

Finally, work out the integral of 1.dx

= xlnx – x + c

Example 2

Work out the integral of 2xln5x.

Like example 1 make u = ln5x and v’ = 2x. Always make u equal to the function with natural log.

Now differentiate u to get u’ = 1/x and integrate v’ to get v = x²

Next substitute these values into the formula for integration by parts.

∫uv’ dx= uv - ∫u’vdx

∫2x.ln5xdx = ln5x.x² - ∫(1/x).x²dx

Tidy this up and (1/x) × x² will simplify to x:

=x²ln5x - ∫x.dx

= xlnx – 0.5x² + c

Let’s look at one last example that involves the integration of natural log x.

Example 3

Work out the integral of x³ln8x,

Like example 1 and 2 use the formula for integration by parts:

Just like examples 1 and 2, make u = ln8x and v’ = x³

Now differentiate u to give u’ = 1/x and integrate v’ to give v = ¼x⁴

Next substitute these values into the formula for integration by parts.

∫uv’ dx= uv - ∫u’vdx

∫x³. ln8xdx = ln8x. ¼x⁴ - ∫(1/x).0.25x⁴dx

Tidy this up and (1/x).0.25x⁴ will simplify to get ¼x³

= ln8x. ¼x⁴ - ∫¼x³dx

= ¼x⁴ln8x – (1/16)x⁴ + c

So that is all you need to know about integration involving natural logs.

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Comments 1 comment

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rahul0324 4 years ago from Gurgaon, India

Calculus is divine... It would be impossible to understand the dimensions of the universe without it...

If Universe is the art.... Calculus is its describing literature..

A very interesting hub.... well explained easy to comprehend explanations..

Great

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