此为 Mathematica 4 文档,内容基于更早版本的 Wolfram 语言
查看最新文档(版本11.1)

Integrate

Usage

Integrate[f, x] 给出不定积分 .
Integrate[f,  x, xmin, xmax ] 给出定积分 .
Integrate[f,  x, xmin, xmax ,  y, ymin, ymax ]给出多重积分  .


Notes

Integrate[f, x] 可以输入为  f  x.
 能输入为 AliasIndicatorintAliasIndicator 或 \[Integral].
 不是普通的 d; 它被输入为 AliasIndicatorddAliasIndicator 或 \[DifferentialD].
Integrate[f,  x, xmin, xmax ] 能用xmin作为  的下限和xmax作为  的上限输入。 •多重积分使用标准迭代记号的变体。给出的第一个变量相应于最外边的积分,在最后进行计算。
Integrate 能计算有理函数的积分。它也能计算涉及指数,对数,三角函数和反三角函数的积分,只要根据相同的函数集合得出结果。
Integrate 能根据许多特殊函数给出结果。
Integrate 对它不能明确进行计算的积分进行一些简化。 • 可以通过运用 N 到一个定积分得到一个数值结果。 • 可以给涉及Integrate的模式分配值给出新类型的积分结果。 • 积分变量可以是任何表达式。然而,Integrate仅使用它的字面形式。例如,对象  并不转换为  .
• 对不定积分,Integrate试图找出对几乎大多数参数值是正确的结果。
• 对定积分,给出下面的选项:
"\!\(\*StyleBox[\"\\\"PrincipalValue\\\"\", \"MR\"]\) ""\!\(\*StyleBox[\"\\\"False\\\"\", \"MR\"]\) "是否找柯西主值
"\!\(\*StyleBox[\"\\\"GenerateConditions\\\"\", \"MR\"]\) ""\!\(\*StyleBox[\"\\\"True\\\"\", \"MR\"]\) "是否产生涉及参数条件的回答
"\!\(\*StyleBox[\"\\\"Assumptions\\\"\", \"MR\"]\) ""\!\(\*StyleBox[\"\\\"{}\\\"\", \"MR\"]\) "假设在参数间关系
Integrate 本质上能计算所有的不定积分和在标准表中列出的大多数定积分。
• 在 StandardForm, Integrate[f, x]输出为  f  x.
• 参见 Mathematica 全书: 1.4.4, 节 1.5.3 and 节 3.5.6.
• 实现注释: 参见节 A.9.5.
Further Examples

Indefinite integrals

Here are three indefinite integrals that are evaluated by the (extended) Risch algorithm.

In[1]:=  

Out[1]=

In[2]:=  

Out[2]=

In[3]:=  

Out[3]=

Here are two indefinite integrals that are evaluated by special table lookup rules.

In[4]:=  

Out[4]=

In[5]:=  

Out[5]=

Definite integrals

Here is a definite integral.

In[6]:=  

Out[6]=

Options: Assumptions and GenerateConditions

When an integrand depends on a parameter, the indefinite integral should be considered valid for "generic" values of the parameter. For certain values the reported integral may be meaningless, as is the case here for  .

In[7]:=  

Out[7]=

By contrast, when you ask for a definite integral, Mathematica tries to return a result that is always valid, if necessary by stating validity conditions.

In[8]:=  

Out[8]=

The option Assumptions lets you explicitly state conditions on the parameters. If you are only interested in real values of  that are less than 0, the reported result is unconditional.

In[9]:=  

Out[9]=

The same unconditional result is returned if you override the default setting of the option GenerateConditions. Unless you know beforehand under what conditions the result is valid, this usage can lead to nonsense.

In[10]:=  

Out[10]=

This integral is a definition of the Beta function.

In[11]:=  

Out[11]=

Here is a more complicated example. For  , the reported indefinite integral is not valid; the true integral is Log[x], since the integrand reduces to  .

In[12]:=  

Out[12]=

In[13]:=  

Out[13]=

Here is the result when you ask for a definite integral.

In[14]:=  

Out[14]=

Even for a definite integral, you should use results with caution if they depend on a parameter. In this example, the result is not valid if  is such that there is a singularity in the segment  (that is, if  is real and at least  ).

In[15]:=  

Out[15]=

In[16]:=  

Out[16]=

Options: PrincipalValue

Sometimes an integrand has a singularity on the path of integration that causes the integral (in the Riemann sense of an area) to diverge. This is the case for this function, at the point  .

In[17]:=  

Out[17]=

The integral may nonetheless exist in the sense of the Cauchy principal value. This value is defined by considering the integral obtained by omitting a small interval centered at the singularity, and taking the limit as the interval's length shrinks to zero.

In[18]:=  

Out[18]=

Cauchy principal values can also be obtained numerically, using the package "NumericalMath`CauchyPrincipalValue`".

In[19]:=  

In[20]:=  

Out[20]=

In[21]:=  

Numerical integration

When a function cannot be integrated symbolically, you can usually obtain a definite integral numerically using NIntegrate.

In[22]:=  

Out[22]=

In[23]:=  

Out[23]=

See the Further Examples for NIntegrate.