The `int`

command is used to compute both definite and indefinite
integrals of Maple expressions. Its syntax is basically

> int(f,x);

where `f`

is an algebraic expression and `x`

is the integration
variable. For example, to compute

d*x* ,

we execute the Maple command

> int((3*x-6)/(x^2-4),x);

Notice that Maple does not provide the constant of integration. You will
occasionally have to take this into account and provide your own constant.

You must specify the variable of integration. In expressions involving other parameters, Maple assumes that you want the integral of the expression as the variable you specify changes and that all other parameters in the expression represent constants:

> int(exp(a*x),x);

To compute a definite integral, the *range* over which the integration
variable moves must be specified:

> int(x^2*exp(x),x=0..2);

The `int`

command in Maple has a particular behavior in certain
situations:

- If Maple cannot compute the integral in
*closed form*, it will return it unevaluated:> int(ln(sin(sqrt(x^12-5*x^7+50*x+2))),x);

- Sometimes the integral cannot be evaluated in closed form in terms
of
*elementary*functions, but the answer has a special name in mathematical circles due to its importance in applications. For example,> int(sin(2*x)/x,x);

Here`Si`

is the*special name*of one of these functions which appear frequently in mathematical physics. To learn more about it, ask Maple for help on it with the command`?Si`

. This will bring up a window with information about the function`Si`

. If Maple responds to an integral with one of these functions, it is quite likely that the integral cannot be evaluated in terms of elementary functions. - Sometimes, a result may be expressed in terms of the roots of a
polynomial which does not factor over the rationals. For example,
> int(1/(x^8 +1),x);

In this answer, the sum is taken over all roots*R*of the polynomial 1 + 16777216*z*^{8}, and the summand is*R*log(*x*+8*R*).

As with `diff`

, there is an *inert* form `Int`

of the integral
command which can be used in combination with `int`

to produce easily
readable worksheets:

> Int(ln(1+3*x), x=1..4);

> Int(ln(1+3*x), x=1..4)=int(ln(1+3*x),x=1..4);

The inert form `Int`

is also very useful in many situations when
you wish to delay the evaluation of an integral, as we shall see
below.

`evalf`

command, you can force Maple to apply a numerical
approximation technique for definite integration:
> evalf(Int(sqrt(1+x^10),x=0..1));

Notice that the we are using the inert form of the integration
command, `Int`

. This prevents Maple from attempting to evaluate
the integral symbolically and then applying `evalf`

to the
answer. In many cases, this can save a huge amount of time, because
Maple will work very hard to try to compute the symbolic form of the
integral. For example, approximating
*e*^{sin(x)} d*x* with the command

> evalf(int(exp(sin(x))),x=0..1);

took more than 10 times the amount of time needed to execute

> evalf(Int(exp(sin(x))),x=0..1);

More complicated integrals can have even more dramatic differences.

`student`

and you must load
this library into computer memory before you can use it:
> with(student):

Once loaded, you can play with it. For example, if you want to use the left sum approximation to an integral, the height of each rectangle is determined by the value of the function at the left side of each interval. You may specify the number of intervals you wish to use. If you do not, Maple will use four intervals by default:

> leftsum(x^4*ln(x),x=1..4,10);

Maple can also draw pictures of the rectangles used to approximate the integral of a function on a given interval:

> leftbox(1/x,x=1..2,10);

The following is the result of using the Simpson's rule for the function
*xsin*(*x*^{2}) on the interval
1 *x* 5. Since the number of intervals
is not specified, Maple assumes four equal intervals by default:

> simpson(x*sin(x^2),x=1..5);

> int(int(x^2*y^3,x=0..y),y=2..3);

This is literally an iterated integral: the integral of *x*^{2}*y*^{3} with respect
to *x* on the interval [0, *y*] is nested inside the integral with respect to
*y* on the interval [2, 3], in effect calculating

2002-08-29