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int

definite and indefinite integration

 

Calling Sequence

Parameters

Description

Examples

Details

Compatibility

Calling Sequence

int(expression,x, options)

∫expressionⅆx

int(expression,x=a..b, options)

∫abexpressionⅆx

int(expression, [x, y, ...], options)

expressionⅆxⅆy

int(expression, [x = a..b, y = c..d, ...], options)

cdabexpressionⅆxⅆy

Parameters

expression

-

algebraic expression; integrand

x, y

-

names; variables of integration

a, b, c, d

-

endpoints of interval on which integral is taken

options

-

(optional) various options to control the type of integration performed. For example, numeric=true will perform numeric instead of symbolic integration. See int/details for more options.

Description

• 

The int(expression, x) calling sequence computes an indefinite integral of the expression with respect to the variable x. Note: No constant of integration appears in the result.

• 

The int(expression, x = a..b) calling sequence computes the definite integral of the expression with respect to the variable x on the interval from a to b.

• 

The int(expression, [ranges or variables]) calling sequence computes the iterated definite integral of the expression with respect to the variables or ranges in the list in the order they appear in the list. Note: The notation int(expression, [x = a..b, y = c..d]) is equivalent to int(int(expression, x = a..b), y = c..d) except that the single call to int accounts for the range of the outer variables (via assumptions) when computing the integration with respect to the inner variables.

• 

You can enter the command int using either the 1-D or 2-D calling sequence.  For example, int(f,x) is equivalent to ∫fⅆx.

• 

If any of the integration limits of a definite integral are floating-point numbers (e.g. 0.0, 1e5 or an expression that evaluates to a float, such as exp(-0.1)), then int computes the integral using numerical methods if possible (see evalf/int). Symbolic integration will be used if the limits are not floating-point numbers unless the numeric=true option is given.

• 

If Maple cannot find a closed form expression for the integral (or the floating-point value for definite integrals with float limits), the function call is returned.

• 

Note: For information on the inert function, Int, see int/details.

Examples

No constant of integration appears in the result for indefinite integrals.

f7x3+3x2+5x:

∫fⅆx

74x4+x3+52x2

(1)

∫sinxⅆx

cosx

(2)

∫xx31ⅆx

16lnx2+x+1+133arctan132x+13+13lnx1

(3)

∫ⅇx2ⅆx

12πerfx

(4)

If Maple cannot find a closed form expression for the integral, the function call is returned.

∫ⅇx2lnxⅆx

∫ⅇx2lnxⅆx

(5)

Compute definite integrals.

∫0πsinxⅆx

2

(6)

∫0∞ⅇx2lnxⅆx

14πγ12πln2

(7)

∫0∞ⅇx2lnx2ⅆx

116π5/2+18γ2π+12γln2π+12ln22π

(8)

An Elliptic integral

∫2312t43t22ⅆt

155EllipticF137,155155EllipticF122,155

(9)

A double integral

intxy2,x,y

16x2y3

(10)

∫22∫0yxy2ⅆxⅆy

325

(11)

If either of the integration limits are floating-point numbers, then int computes the integral using numerical methods.

∫2.02∫0.0yxy2ⅆxⅆy

6.400000000

(12)

An integral with decimal limits using numerical methods:

∫0.751.25xx3+1ⅆx

0.2459707569

(13)

To apply symbolic integration methods instead, use numeric=false:

intxx3+1,x=0.75..1.25,numeric=false

133arctan16316ln13+12ln7+133arctan12312ln3

(14)

The option numeric=true or simply numeric may also be used to compute a numerical integral even with exact limits:

intxx3+1,x=34..54,numeric

0.2459707569

(15)

Details

  

For detailed information including:

• 

Integration over a complex interval

• 

Numerical integration

• 

Handling discontinuities

• 

Series expansions

• 

Inert form of the int command, Int

  

see the int/details help page.

Compatibility

• 

The int command was updated in Maple 2016; see Advanced Math.

See Also

convert/rational

diff

evalf/int

int/details

IntegrationTools

VectorCalculus

VectorCalculus[int]

 


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