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Scilab Help >> Polynomials > polyint

# polyint

Polynomial integration

### Syntax

```polyint // example
Q = polyint(P)
Q = polyint(P, Const)```

### Arguments

P, Q
arrays (from scalars to hypermatrices) of polynomials with real or complex coefficients, of same size.

Const
Set of real or complex constants of integration. `length(Const)` sets the number of consecutive integrations to perform. By default, 0 is used. `Const` can be one of the following:
• a scalar: then only the first antiderivatives are computed, and all use the same `Const` value.
• an array of size size(P): then only the first antiderivatives are computed, with the respective `Const(i)` for `Q(i)`.
• a list that can mix scalars, arrays of size size(P), or undefined elements. Then `Const(i)` is used for the ith antiderivatives. Undefined elements are equivalent to 0.

### Description

polyint(..) computes the first or any nth antiderivative of each polynomial of the input array.

### Examples

Single integration (first order):

```s = %s;
P = [-2, s ; s^2-3, s-2*s^3]
polyint(P)
polyint(P, 1)
polyint(P, [1 2 ; 3 4])```
```--> P = [-2, s ; s^2-3, s-2*s^3]
P  =
-2      s
-3 +s²  s -2s³

--> polyint(P)
ans  =
-2s               0.5s²
-3s +0.3333333s³  0.5s² -0.5s⁴

--> polyint(P, 1)
ans  =
1 -2s               1 +0.5s²
1 -3s +0.3333333s³  1 +0.5s² -0.5s⁴

--> polyint(P, [1 2 ; 3 4])
ans  =
1 -2s               2 +0.5s²
3 -3s +0.3333333s³  4 +0.5s² -0.5s⁴
```

Multiple integrations:

```s = %s;
polyint(1-3*s, list(-1,1))
P = [3 ; 1-s ; 6*s^2-2]
polyint(P, list(2, [-2 1 0]))
polyint(P, list(, [-2 1 0]))

k = list(); k(5) = 0;
polyint(1+0*s, k)
1 / factorial(5)```
```--> polyint(1-3*s, list(-1,1))
ans  =
1 -s +0.5s² -0.5s³

--> P = [3 ; 1-s ; 6*s^2-2]
P  =
3
1 -s
-2 +6s²

--> polyint(P, list(2, [-2 1 0]))
ans  =
-2 +2s +1.5s²
1 +2s +0.5s² -0.1666667s³
2s -s² +0.5s⁴

--> polyint(P, list(, [-2 1 0]))
ans  =
-2 +1.5s²
1 +0.5s² -0.1666667s³
-s² +0.5s⁴

--> k = list(); k(5) = 0;
--> polyint(1+0*s, k)
ans  =
0.0083333s⁵

--> 1 / factorial(5)
ans  =
0.0083333
```

• derivat — derivative of polynomials or of rationals

### History

 Version Description 6.1.1 polyint() introduced.