Here is an example of a definite integral body of a rational fraction:
Define the definite integral of the rational fraction of a rational function, i.e., for the rational function $f(x)= frac$, where $p(x)$ and $q(x)$ are one-time polynomials, and we can calculate their definite integrals by the following steps:
1.Write $f(x)$ in the standard form, i.e., $f(x)= frac$, where $a$ and $b$ are constants.
2.Calculate the indefinite integral of $f(x)$ over the interval $[a,b]$ and get $ int f(x)dx= int fracdx=a ln|x+b|+c$, where $c$ is constant.
3.Calculate the definite integral of $f(x)$ over the interval $[a,b]$ and get $ int a bf(x)dx= int a b fracdx=a ln|x+b||_a^b=a\ln\frac-a\ln\frac=a\ln\frac$。
Through the above steps, we can get the formula for calculating the definite integral of a rational fraction, i.e., $ int a bf(x)dx=a ln frac$, where $f(x)= frac$ is a rational function, and $p(x)$ and $q(x)$ are primary polynomials.
The definite integral of rational fractions has important application value in mathematical analysis. For example, when solving some physics problems, we need to calculate definite integrals to solve problems such as the trajectory of an object, heat conduction, and so on. In addition, the definite integral of a rational fraction is also the basis for series expansion, so it plays an important role in mathematical analysis.
When calculating the definite integral of a rational fraction, we need to choose the appropriate method according to the actual situation of the function. For example, if the function is a simple function such as a polynomial function or a trigonometric function, we can directly use the corresponding formula to calculate itIf the function is a rational function, we can write it as a rational fraction for calculation. If the function is complex or cannot be solved directly, we can use numerical or analytical methods to approximate it.
In short, the definite integral of rational fractions is one of the important concepts in mathematical analysis, and it has a wide range of applications. By mastering its definition and calculation methods, we can better solve various mathematical problems as well as practical application problems.