Mathematica 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:
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For the definite integral
, the second argument is a list of the form
:
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To do the multiple integral
, use a mix of a variable and a range:
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Alternatively, you can use
Integrate twice:
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Calculating the area of a circle is a classic calculus problem. An intuitive way to approach this is the integral
, which involves substitution:
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Integrate gives exact answers to many improper integrals; for example,
:
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Suppose that there is no closed form for a definite integral; for example,
:
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In that case, you can get an approximation with
NIntegrate:
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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:
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Repeating the calculations is fast because of caching:
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NIntegrate can also compute multiple integrals:
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