The ∫() Command

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Command Summary

Takes the integral of an expression.

Command Syntax

∫(expression,variable)
∫(expression,variable,constant)
∫(expression,variable,lower,upper)

Menu Location

Press [2nd][7] to enter ∫(.

Calculator Compatibility

This command works on all calculators.

Token Size

2 bytes

∫(expression,variable) takes the integral of expression (symbolically) with respect to variable. All other variables are treated as constant.

There are three ways to use ∫(). The syntax above returns an indefinite integral. ∫(expression,variable,c) does the same, but with a constant of integration, c (this will just get added on to the result). Finally, ∫(expression,variable,a,b) takes a definite integral from a to b. These limits can be anything, including undefined variables, ∞ and -∞, as long as they don't depend on variable.

:∫(x^2,x)
           x^3/3
:∫(x^2,x,c)
           x^3/3+c
:∫(x^2,x,a,b)
           b^3/3-a^3/3

Indefinite integrals are always computed exactly or not at all: if a part of the expression (or the entire expression) can't be integrated, the result will stay in terms of ∫(). However, definite integrals will sometimes be approximated, depending on the Exact/Approx mode setting:

• If EXACT, integrals will never be approximated.
• If AUTO, the calculator will approximate integrals like ∫(e^(-x^2),x,-1,1) that it can't compute exactly.
• If APPROX, all definite integrals will be done numerically if possible.

:∫(e^(-x^2),x)
           ∫(e^(-x^2),x)
:∫(e^(-x^2),x,-1,1)
           2*∫(e^(-x^2),x,0,1) (in EXACT mode)
           1.49365 (in AUTO or APPROX mode)

Finally, you can take multiple integrals by applying ∫() to the result of another ∫() (any number of times). The integration limits of the inner integrals can involve the variables of the outer integrals.

:∫(∫(x*y,x),y)
           y^2*x^2/4
:∫(∫(x*y,x,0,y),y,0,1)
           1/8

If the expression is a list or matrix, ∫() takes the integral of each element.

Error Conditions

140 - Argument must be a variable name happens when the variable of integration isn't a variable.

220 - Dependent limit happens when the integration limits depend on the variable of integration.

Related Commands

• d() (derivative)
• nInt()
• ∑() (sum)