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


Today is the ninth article in our advanced mathematics series, and we continue to look at antiderivatives.

In the last article, we reviewed the definition and simple properties of an antiderivative. We can simply think of an antiderivative as the inverse operation of the derivative. So what we’re going to do is we’re going to take the derivative, and we’re going to reverse it.

In addition to the basic definitions, we have introduced some simple properties and a table of points for common integrals. However, it is still very difficult to solve the integral for many complex functions according to the existing properties, so the focus of this article is to continue to introduce the operation properties of indefinite integrals, so as to simplify the calculation process of some complex functions. Or even perform calculations that would otherwise be impossible. Today’s introduction is the most commonly used substitution integral method.

Substitution method is often used in mathematics. We often use substitution method to reduce the difficulty of problems, whether it is derivation calculation or some complex functions. Similarly, in the antiderivative solution, we can also use the substitution method to carry out. Usually there are two categories of substitution, why are there two categories? What’s the difference between the two categories? These questions can be put aside for a while and will become clear after reading the article.


Substitution method of the first kind


The first kind of substitution is easier to understand, and is actually the reverse of the chain rule.

For example, we have functions, obviously function F(u) is the original function of F(u), so:

If u is an intermediate variable, and, we haveDerivative, according to the chain derivative rule of composite function, it can be obtained:


We can obtain the substitution formula of the antiderivative by integrating the above expression in reverse:


So we got the formula by simple derivation, so how does this formula work? At first glance, it’s a little hard to do, and that’s normal, but we need to keep simplifying.

So let’s say we want to figure outIt’s a little bit more difficult to solve it directly, if we can somehow convert g of x to PI, then we can apply the formula to get:


And then the integral of g of x becomes the integral of f of u, and if we can find the antiderivative of f of u, then we have the antiderivative of g of x. In general, the function f(u) is much simpler than the original function g(x), and that’s what the substitution method means.

Let’s take a look at an example:


Because the x in the denominator has a coefficient, we can’t use the integral formula directly. And then we have to make a substitution, and it’s not hard to imagine that we could use u is equal to 3 plus 2x. Since we want to solve for f(u)du, we find that the derivative of u with respect to x is 2, so we can transform the equation:


And what we can see from this example is that the essence of the substitution method is very simple, that after the substitution, we have to make up f(u)du. And when we get there, we can use u as a variable.

Let’s look at a more complicated example:


In this case, the function we’re going to calculate is a little bit more complicated,It involves both trigonometry and squaring. It’s not gonna work. We need to do it firstAs a, so we can apply the formula of integration and difference, and get:

It’s much easier here:


If u = 2x, the above equation can be transformed into:



The second kind of substitution method


Now that we are familiar with the first type of substitution, let’s look at the second type of substitution.

In the first type of substitution method, we replace a relatively complicated function with a new variable. For example, we replace 2x or 2x+3 with u, which simplifies the subsequent operations. The second kind of substitution method takes the opposite approach, transforming a single variable into a complex expression. For example, we use trigonometric functions or polar coordinates to represent the original x, which is often common in high school math problems, especially analytical geometry problems. We often set up polar coordinates, using polar formulas to simplify the calculation.

In other words, the second substitution method is just the opposite of the logic of the first substitution method, which is to convert x to x. So the substitution formula is:


But if you want to do that, if f of x is integrable it means that the integral must exist, but the substitution on the right-hand side does not. So we need assurancesThe antiderivative of phi exists. And secondly, after we’ve done our substitution, we need to useInverse function of phiPlug it back in. But to make sure that the inverse exists and is differentiable, we can simply think of the antiderivativeIt’s monotonically differentiable on some interval, and.

We write the substitution formula according to the above definition:


We can also use itThe chain ruleTo prove that, let’s sayThe antiderivative of phi is theta, so, we haveTaking the derivative, we can get:


Let’s also look at an example:

In this case we have a square root, and we have a square term, and that looks very cumbersome, and that’s when we need to make a substitution. because, so we can make.. We can substitute into the original formula, and get:


In fact, this is the second example we discussed above, and we calculated the answer earlier:, we can substitute into the original formula, and get:


Due to the, so.., we can get the final result by substituting these into the above equation:


This concludes the two substitution methods, which seem simple, but we can derive many different uses from the common integration formulas we introduced earlier. But want to understand all these uses need us to be very familiar with the integral formula and substitution application, these are not one or two articles can do, must do a lot of practice, I think the postgraduate students should have a very deep experience.

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.