Takes the integral of an expression.

∫(*expression*,*variable*)

∫(*expression*,*variable*,*constant*)

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

**Menu Location**

Press [2nd][7] to enter ∫(.

This command works on all calculators.

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.