This article originated from personal public account: TechFlow, original is not easy, for attention


Today is the tenth article in our series on advanced mathematics.

Today we’re going to look at another way of solving antiderivatives, integration by parts, which is very common, even more common than substitution. In my few memories of high numbers, this was one of the required tests.

Although this is very important, it is not difficult and the derivation is very simple, so this article is hardly difficult and there is no formula derivation.


Principle and Derivation

The principle of integration by parts is very simple, in fact, also derived from the derivation of the derivative formula. When we introduced antiderivatives, we introduced a simple formula for integrating by adding and subtracting functions. The formula of integration by parts this time comes from the derivation rule of the product of two functions.

Let’s say u and v are functions of x, and their derivatives are continuous. According to the derivation formula, we can get the derivative formula of the product of the function UV:


This formula should be simple, we are familiar with high school math, and then we do a simple transposition, we get:


And then we take the antiderivative of both sides of the equation:


The above formula can also be simplified as:


So this is the derivation of our integration by parts formula, isn’t it easy? Some of you might be a little confused about this, like why the integral of uv prime becomes UV. And the reason for that is very simple, because when you take an antiderivative, you take the derivative of a function and you integrate it, you get the function itself. And that’s why maybe the integral was the inverse of the derivative.

At some point, we want to askIt’s not so easy to findThat’s easy. That’s when we can use the integration by parts formula to get the answer.


Choice of u and v

In integration by parts, the most important thing is the choice of u and v, which will directly affect our calculation and complexity. Let’s look at the following example.

Let’s say we want to figure out, this formula is more troublesome, both the first type of substitution method and the second type of substitution method are not easy to solve. Let’s try integration by parts. There are only two parts in this formula, which is kind of obvious. Let’s say, then, we substitute integration by parts:


It’s easy to get the original function, so the overall answer is:


But why does it have to be? If we makeHow about that?

Of course I could, butThe whole thing is going to be a lot of troubleWe can just plug it in and see if, then., we can get a complicated formula:


It’s probably harder to integrate this than it was before, but one of the things that this example shows you is that you can’t just blindly choose u and v, you can’t just pick a function and simplify the calculation.

In general, there are two rules that we can use to make sure that we get as good a result as possible by integration by parts, and the first rule isV is easy to calculate. In the example we just did, if dv was complicated, it would make our v complicated. It’s going to make the VDU a little bit harder to calculate. The second principle isthanIt’s easy to calculate, and this is obvious, because otherwise why would we use integration by parts? Let’s just do it.


A little trick

Well, we can see a little bit from the example above and from the formula for integration by parts, the premise of integration by parts is to make v as easy as possible, what kind of functions are easy to integrate and take derivatives of?

Obviously, trigonometric functions and all kinds of functions where e appears. So integration by parts is preferred for functions with trigonometric functions and natural base e.

Let’s look at another example:


That’s what happens in this exampleWe knowIt’s a good thing,Its integral and derivative are equal to itselfWell, it’s perfect for v. So let’s say, so, we can get the answer by substituting it into the formula:


Let’s look at another example:


We make, so, can be obtained by substituting:


In addition to the trick in function selection, another trick is that our integration by parts method can be used for multiple times. For some complicated formulas, one resolution is not enough. At this time, we can consider continuing to use integration by parts for multiple resolution. Let’s look at an example:


Just like before, let’s make, so. We can substitute into the original formula, and get:


And we see that there’s an integral on the right hand side that’s not very easy to integrate, so we’re going to use integration by parts, then, we can get:



With substitution

Integration by parts can be divided multiple times, and another killer is that it can also be used in conjunction with substitution methods. The combination of these two methods increases the power of the formula, further expanding the range of applications.

Let’s look at an example:


Even though e appears in this formula, its exponent is also a function, and it’s not easy to use integration by parts. And this is where we need to combine substitution, let’s say, so

We can get:


We are already familiar with this formula, but using integration by parts, we can easily get:



conclusion

So that’s the end of integration by parts, not only integration by parts, but the whole method of solving antiderivatives. In fact, there are only two methods, substitution method and integration by parts, although these two methods are simple, but if you use skillfully, the power is not small, can solve many seemingly intractable integration problems. You can watch the two articles together.

Today’s article is all of these, if you feel that there is a harvest, please feel free to point attention or forward, your help is very important to me.