Mathematica 9 is now available
SEARCH MATHEMATICA 8 DOCUMENTATION
THIS IS DOCUMENTATION FOR AN OBSOLETE PRODUCT.
SEE THE DOCUMENTATION CENTER FOR THE LATEST INFORMATION.
Mathematica > Calculus >
Built-in Mathematica SymbolTutorials »|See Also »|More About »

Integrate ()

Integrate[f, x]
gives the indefinite integral ∫f dx.
Integrate[f, {x, xmin, xmax}]
gives the definite integral .
Integrate[f, {x, xmin, xmax}, {y, ymin, ymax}, ...]
gives the multiple integral .
  • Integrate[f, {x, xmin, xmax}] can be entered with xmin as a subscript and xmax as a superscript to .
  • Multiple integrals use a variant of the standard iterator notation. The first variable given corresponds to the outermost integral, and is done last.  »
  • Integrate can evaluate integrals of rational functions. It can also evaluate integrals that involve exponential, logarithmic, trigonometric and inverse trigonometric functions, so long as the result comes out in terms of the same set of functions.
  • Integrate can give results in terms of many special functions.
  • Integrate carries out some simplifications on integrals it cannot explicitly do.
  • You can get a numerical result by applying N to a definite integral.  »
  • You can assign values to patterns involving Integrate to give results for new classes of integrals.
  • The integration variable can be a construct such as x[i], or any expression whose head is not a mathematical function.
  • For indefinite integrals, Integrate tries to find results that are correct for almost all values of parameters.
  • For definite integrals, the following options can be given:
Assumptions$Assumptionsassumptions to make about parameters
GenerateConditionsAutomaticwhether to generate answers that involve conditions on parameters
PrincipalValueFalsewhether to find Cauchy principal values
  • Integrate can evaluate essentially all indefinite integrals and most definite integrals listed in standard books of tables.
EXAMPLESCLOSE ALL
Basic Examples   (6)
Indefinite integral:
In[1]:=
Click for copyable input
Out[1]=
 
Definite integral:
In[1]:=
Click for copyable input
Out[1]=
 
Use Esc int Esc to enter and Esc dd Esc to enter :
In[1]:=
Click for copyable input
Out[1]=
In[2]:=
Click for copyable input
Out[2]//TraditionalForm=
 
Use Ctrl+_ to enter the lower limit, then Ctrl+% for the upper limit:
In[1]:=
Click for copyable input
Out[1]=
 
Multiple integral with x integration outermost:
In[1]:=
Click for copyable input
Out[1]=
In[2]:=
Click for copyable input
Out[2]=
 
Integrals that may not converge are by default returned as conditionals:
In[1]:=
Click for copyable input
Out[1]=
New in 1 | Last modified in 6
© 2013 Wolfram Research, Inc. Japanese
Ask a question about this page  |  Suggest an improvement  |  Leave a message for the team