Basic integration formulas, Linear substitution.
We start looking for primitive functions by remembering the derivatives of a few important functions.
The first one is xn
Differentiation decreases the exponent by 1. Integration increases it by 1.
Although it's a little troublesome, if
But here comes the solution.
Then, finally an anchor point in our lives.
The list is going to be fairly long.
And this is only the beginning.
And now, we have to clarify a few very important things.
Here is one of them:
And here is the other one:
Let's try to figure out this one:
It seems logical that
Only there is a little trouble.
Integration is the reverse of differentiation, so if we integrate a function and then differentiate it, we should get back exactly to the original function.
But that doesn't seem to be true here.
We don't arrive back to the original function, because differentiation inserts this multiplier of 3 here.
But this can be helped.
If the exponent is some sort of ax+b type polynomial,
then when we integrate, we have to multiply by .
Let's see this one, for instance:
Here, instead of being in the exponent, the ax+b polynomial is in the denominator.
We will need this method quite often, so we should store it in a handy place in our head.
Now comes the real thrill!