Integration

Here is the integral ∫(x^n) dx in the Wolfram Language:

Integrate[x^n,x]

Here is a slightly more complicated example:

Integrate[1/(x^4-a^4),x]

The Wolfram Language knows how to do almost any integral that can be done in terms of standard mathematical functions. But you should realize that even though an integrand may contain only fairly simple functions, its integral may involve much more complicated functions—or may not be expressible at all in terms of standard mathematical functions.

Here is a fairly straightforward integral:

Integrate[Log[1-x^2],x]

This integral can be done only in terms of a dilogarithm function:

Integrate[Log[1-x^2]/x,x]

This integral involves (RefLink[Erf, paclet:ref/Erf]):

Integrate[Exp[1-x^2],x]

And this one involves a Fresnel function:

Integrate[Sin[x^2],x]

Even this integral requires a hypergeometric function:

Integrate[(1-x^2)^n,x]

This integral simply cannot be done in terms of standard mathematical functions. As a result, the Wolfram Language just leaves it undone:

Integrate[x^x,x]

Integration.

Here is the definite integral (∫_a^b)(sin^2)(x) dx:

Integrate[Sin[x]^2,{x,a,b}]

Here is another definite integral:

Integrate[Exp[-x^2],{x,0,Infinity}]

The Wolfram Language cannot give you a formula for this definite integral:

Integrate[x^x,{x,0,1}]

You can still get a numerical result, though:

N[%]

This evaluates the multiple integral (∫_0^1)dx(∫_0^x)dy ((x^2)+(y^2)). The range of the outermost integration variable appears first:

Integrate[x^2+y^2,{x,0,1},{y,0,x}]

This integrates (x^10) over a circular region:

Integrate[x^10 Boole[x^2+y^2<=1],{x,-1,1},{y,-1,1}]