Composite function, The S4 integration rule, Special cases, Integrate the product by parts, Partial integration, Integration by parts and S4 step by step, Cases with reverse casting, Reduction formula.
This theorem is really about integrating composite functions. But unfortunately composite functions are a bit problematic, as computing their integral is usually a quite hopeless undertaking.
There is no elementary primitive function for any of these functions:
Therefore sadly, we cannot compute these integrals. I don't mean today, but at all.
We may have some hope, if these functions are multiplied by the derivative of their internal functions.
It is worth to remember a few special cases:
Here are a few exercises for them.
There are cases where we have to invest some energy to get everything right.
There are usually two possibilities.
The easier case is when the function we need to integrate differs from the ideal situation only by a constant. The other case is when the factors containing x are also different.
If the difference is only in a constant, then it can be resolved easily:
The other case is much more unpleasant. Let's see an example for that!
At first sight, it looks like an
type case, but there is a little trouble.
Here, we need to have the derivative of the exponent, but the derivative of x2 is 2x.
Here comes an idea: if we need 2x there, then well, let's write 2x there.
But that would change the problem. To prevent the multiplication from changing the problem, we should also divide by 2x.
So, we multiplied by 2x, but also divided by 2x, thus the original problem did not change.
On the other hand, here appeared the derivative of the exponent, so now we can integrate.
The only question is what to do with this part.
We will integration by parts.
A bit more integration here, and then it's done.