1. The indefinite integral of an elementary function is not necessarily an elementary function
2. The indefinite integral of a rational function must be an elementary function and can be integrated
3. Some irrational functions can be transformed into rational functions through variable substitution
So far, we can solve indefinite integrals by using linear properties , integrals with substitutions and integrals by parts , but not all indefinite integrals can be solved. For example, the following indefinite integral cannot be solved:
Naturally, we want to investigate what forms of indefinite integrals can be solved. We know that indefinite integral and derivation are mutually inverse operations, because the derivatives of basic elementary functions can be obtained, and elementary functions are obtained by four arithmetic operations and composite operations of basic elementary functions, so elementary functions can also be derived of:
We want to find out what kind of elementary function F(x) is, and its indefinite integral can be obtained.
1
The indefinite integral of a rational function can be solved by the following theorems:
First, we analyze the denominator part q(x) of the rational function R(x) , by the Fundamental Theorem of Algebra:
The final relation can be expanded in terms of the product of the roots:
Next, we hope to simplify R(x) by radicals. First, for real roots, we have:
The red box needs to explain why λ is not equal to 0, because if λ is 0, then P(α) is 0, then α is a real root of P(x), which must be reduced by the denominator q(x), which is the same as the question Contradictory assumptions. If you continue to do this, you can exhaustively decompose all the real roots. Next is the imaginary root:
To sum up, we can divide rational functions into partial fractions:
We use formulas and recursive formulas to solve the above indefinite integral theoretically:
2
Let's give an example to illustrate.
Example 1 (undetermined coefficient method)
Example 2
3
There are some irrational functions that can be replaced with rational functions by variable substitution:
Give a few examples.
Example 3
Example 4
example 5
