# polyint

Polynomial integration

## Syntax

``q = polyint(p,k)``
``q = polyint(p)``

## Description

example

````q = polyint(p,k)` returns the integral of the polynomial represented by the coefficients in `p` using a constant of integration `k`.```

example

````q = polyint(p)` assumes a constant of integration `k = 0`.```

## Examples

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Evaluate the definite integral

`$I={\int }_{-1}^{3}\left(3{x}^{4}-4{x}^{2}+10x-25\right)dx.$`

Create a vector to represent the polynomial integrand $3{x}^{4}-4{x}^{2}+10x-25$. The ${\mathit{x}}^{3}$ term is absent and thus has a coefficient of 0.

`p = [3 0 -4 10 -25];`

Use `polyint` to integrate the polynomial using a constant of integration equal to `0`.

`q = polyint(p)`
```q = 1×6 0.6000 0 -1.3333 5.0000 -25.0000 0 ```

Find the value of the integral by evaluating `q` at the limits of integration.

```a = -1; b = 3; I = diff(polyval(q,[a b]))```
```I = 49.0667 ```

Evaluate

`$I={\int }_{0}^{2}\left({x}^{5}-{x}^{3}+1\right)\left({x}^{2}+1\right)dx$`

Create vectors to represent the polynomials $p\left(x\right)={x}^{5}-{x}^{3}+1$ and $v\left(x\right)={x}^{2}+1$.

```p = [1 0 -1 0 0 1]; v = [1 0 1];```

Multiply the polynomials and integrate the resulting expression using a constant of integration `k = 3`.

```k = 3; q = polyint(conv(p,v),k)```
```q = 1×9 0.1250 0 0 0 -0.2500 0.3333 0 1.0000 3.0000 ```

Find the value of `I` by evaluating `q` at the limits of integration.

```a = 0; b = 2; I = diff(polyval(q,[a b]))```
```I = 32.6667 ```

## Input Arguments

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Polynomial coefficients, specified as a vector. For example, the vector `[1 0 1]` represents the polynomial ${x}^{2}+1$, and the vector `[3.13 -2.21 5.99]` represents the polynomial $3.13{x}^{2}-2.21x+5.99$.

Data Types: `single` | `double`
Complex Number Support: Yes

Constant of integration, specified as a numeric scalar.

Example: `polyint([1 0 0],3)`

Data Types: `single` | `double`
Complex Number Support: Yes

## Output Arguments

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Integrated polynomial coefficients, returned as a row vector. For more information, see Create and Evaluate Polynomials.