how would I go about integrating:
[x^2 / (1+x^2)] dx
Watch carefully:
by definition.
We can also see that
.
This comes from a substitution.
Let so that .
So
.
BUT you are trying to find .
We DO NOT have a formula for that yet because it is not of the form . But you could TRY a substitution.
Watch what happens.
Let .
We can see that .
Can we change into a function that is of the form ? NO, we can not.
So the rule for DOES NOT WORK in this case.
Now see my post regarding trigonometric substitution to show you the method that DOES work.
Alternatively, let's try differentiating and seeing if we get . If we do, then .
.
But we know that and we also know that
So
.
So .
Therefore .
No more arguments please!