∫(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.
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.