Up Learn – A Level MATHs (Edexcel) – Calculus III
Integral of ln(x)
To find the integral of ln(x), use integration by parts. We came up with a game – the ‘one up, one down’ game – to show you exactly how to use integration by parts in all cases.
Last time, we saw that, when we integrate by parts, we’re really using…
When we integrate by parts, we’re really using the product rule for differentiation.
Now, when we’re playing the one up, one down game, we normally differentiate the algebraic part.
Except when the other part is logarithmic. [xln x ]
And that’s because, if we differentiate this part [x], we have to integrate this part [ln x ]
Which, actually, we haven’t seen how to do yet…
But now, ironically, integration by parts allows us to find the integral of this function [ln x dx]
Now, looking at this expression, it doesn’t look like there are even two parts…
But, actually, we could say that the two parts are…
We could say that the two parts are ln x … and, a secret one! [reveal the secrets… 1⋅ln x ]
And so now, given these two parts, which should we differentiate, and which should we integrate?
As usual, we should differentiate the log function, and integrate our secret 1 – as, whenever we have a log function, we always differentiate that!
And so now, find the integral of ln x …
First, we rewrite with integration by parts, like this [xln x -∫x⋅1xdx]
Then, this simplifies to just… 1… [1dx]
Giving us this as our final integral [ln x dx≡xln x -x+c]
And so this time, integrate this [4ln x dx]
First, [click 1 time] we can rewrite this integral like this [4xln x -4x⋅1xdx]
Which simplifies to this… [becomes 4 on right]
Finally, that gives us this integral [4ln x dx≡4xln x -4x+c]
So, to sum up, it’s possible to integrate natural log functions using integration by parts…
For example, to integrate this [ln x ]…
We differentiate the log function, and integrate…
And integrate a secret one!
Then, we just use integration by parts as normal!