# How to | Do an Integral

The Wolfram Language contains a very powerful system of integration. It can do almost any integral that can be done in terms of standard mathematical functions.

To compute the indefinite integral , use Integrate. The first argument is the function and the second argument is the variable:

 In[1]:=
 Out[1]=

For the definite integral , the second argument is a list of the form {variable,lower limit,upper limit}:

 In[2]:=
 Out[2]=

To do the multiple integral , use a mix of a variable and a range:

 In[3]:=
 Out[3]=

Alternatively, you can use Integrate twice:

 In[4]:=
 Out[4]=

Calculating the area of a circle is a classic calculus problem. An intuitive way to approach this is the integral , which involves substitution:

 In[1]:=
 Out[1]=

Integrate gives exact answers to many improper integrals; for example, :

 In[2]:=
 Out[2]=

Suppose that there is no closed form for a definite integral; for example, :

 In[3]:=
 Out[3]=

In that case, you can get an approximation with NIntegrate:

 In[4]:=
 Out[4]=

If you want a numerical result from the start, it is faster to use NIntegrate than to use Integrate and follow it with N.

This compares the time taken for the two methods:

 In[5]:=
 Out[5]=
 In[6]:=
 Out[6]=

Repeating the calculations is fast because of caching:

 In[7]:=
 Out[7]=
 In[8]:=
 Out[8]=

NIntegrate can also compute multiple integrals:

 In[9]:=
 Out[9]=