Ever have an equation you want to derive, but couldn't because you plain don't know how? Well than this is the guide for you!
As a side note, this is actually covered under the Methods study design and a question like this will most probably be asked on either of the two exams.
So without further Apu (hehe, Simpsons related joke):
TRINON'S GUIDE TO ANTI-DERIVATIVES THROUGH DERIVATIVESI'm only going to run through the fundamental method, because there isn't much else to it. It only starts getting hard in Specialist Maths when they start throwing things like differentiate
)
and hence anti-differentiate
)
and things like that.
We start off with an equation that we can't anti differentiate with any method that has been covered in the methods study design.
)
We first multiply this equation by
so that we get
)
.
Next we find the derivative via the product rule:
) = log_e(x) + \frac{x}{x} = log_e(x) + 1)
Next we re-arrange the new equation:
 = \frac{d}{dx}(xlog_e(x)) - 1)
If we now anti-derive both sides we get:
 dx = \int\frac{d}{dx}(xlog_e(x)) dx - \int1 dx)
 dx = xlog_e(x) - x + C)
Now you can Anti-derive the in-anti-derivable!Hope this helps guys. If you've got any questions just ask.
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