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

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

 In:= Out= Alternatively, you can use Integrate twice:

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

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

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

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

 In:= Out= 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:= Out= In:= Out= Repeating the calculations is fast because of caching:

 In:= Out= In:= Out= NIntegrate can also compute multiple integrals:

 In:= Out= 