# Use formula

polynomial 多项式

Basically, there are two types of indefinite integral.

One is `polynomial`, which composed by $$+$$ and $$-$$

Another is `composite function`, which composed by $$\times$$ and $$\div$$

For the first type of integral, we use basic formula to each part of it directly:

$$
\begin{align\*}
&\int \frac{3x^2 - 2x + 1}{x} dx
\ \\
\=& \int (3x - 2 + \frac{1}{x}) dx
\ \\
\=& \frac{3}{2}x^2 - 2x + \ln{|x|} + C
\end{align\*}
$$

For the second one, oh my god, it's very complicate.

Here's a basic template:

$$
\int f(g(x)) \cdot g^\prime(x)dx
\ \\
\int f(g(x)) \cdot dg(x)
\ \\
\int f(u) \cdot du
$$

Let's do some exercise:

$$
\begin{align\*}
&\int xe^{-x^2} \cdot dx
\ \\
\=& \int e^{-x^2} \cdot x dx    &\text{//composite function in front}
\ \\
\=& (-\frac{1}{2}) \int e^{-x^2} \cdot (-2)x dx    &\text{//find a way to make x = } (-x^2)^\prime = -2x \text{ while keeping equation's balance}
\ \\
\=& (-\frac{1}{2}) \int e^{-x^2} \cdot d(-x^2)    &\text{//we knew }g^\prime(x)dx = dg(x)
\ \\
\=& (-\frac{1}{2}) e^{-x^2} + C    &\text{//see } -x^2 \text{ as a part, then apply basic integral formula}
\end{align\*}
$$

$$
\begin{align\*}
\ \ \\
&\int \cos(1-3x) \cdot dx
\ \\
\=&\int \cos(1-3x) \cdot 1dx    &\text{//don't forget 1}
\ \\
\=& (-\frac{1}{3})\int \cos(1-3x) \cdot (-3)1dx    &\text{//find a way to make 1 to -3 while keeping equation's balance}
\ \\
\=& (-\frac{1}{3})\int \cos(1-3x) \cdot d(1-3x)    &\text{//} \frac{dy}{dx} dx = dy
\ \\
\=& (-\frac{1}{3}) \sin(1-3x) + C    &\text{//see 1-3x as a part, then use basic formula}
\ \ \\
\end{align\*}
$$

From these exercises, we know a criticle principle is: $$x$$ of $$dx$$ is the original function, it can be just a $$x$$, it also can be, for example, $$x^2$$.

$$
\begin{align\*}
\int (x) dx^2 = \int ((x^2)^\prime \cdot x) dx = \int (2x \cdot x) dx = \int (2x^2) dx
\end{align\*}
$$

$$
\begin{align\*}
\\
&\int (1 \cdot x) dx    &\text{where's 1 come from? } x^\prime = 1
\end{align\*}
$$


---

# Agent Instructions: Querying This Documentation

If you need additional information that is not directly available in this page, you can query the documentation dynamically by asking a question.

Perform an HTTP GET request on the current page URL with the `ask` query parameter:

```
GET https://yingshaoxo.gitbook.io/university-notes/high-level-math/integration/indefinite-integral/use-formula.md?ask=<question>
```

The question should be specific, self-contained, and written in natural language.
The response will contain a direct answer to the question and relevant excerpts and sources from the documentation.

Use this mechanism when the answer is not explicitly present in the current page, you need clarification or additional context, or you want to retrieve related documentation sections.