# Analytic Solution to Integral of Polynomial

This example shows how to use the `polyint` function to integrate polynomial expressions analytically. Use this function to evaluate indefinite integral expressions of polynomials.

### Define the Problem

Consider the real-valued indefinite integral,

`$\int \left(4{x}^{5}-2{x}^{3}+x+4\right)\phantom{\rule{0.2222222222222222em}{0ex}}dx$`

The integrand is a polynomial, and the analytic solution is

`$\frac{2}{3}{x}^{6}-\frac{1}{2}{x}^{4}+\frac{1}{2}{x}^{2}+4x+k$`

where $k$ is the constant of integration. Since the limits of integration are unspecified, the `integral` function family is not well-suited to solving this problem.

### Express the Polynomial with a Vector

Create a vector whose elements represent the coefficients for each descending power of x.

`p = [4 0 -2 0 1 4];`

### Integrate the Polynomial Analytically

Integrate the polynomial analytically using the `polyint` function. Specify the constant of integration with the second input argument.

```k = 2; I = polyint(p,k)```
```I = 1×7 0.6667 0 -0.5000 0 0.5000 4.0000 2.0000 ```

The output is a vector of coefficients for descending powers of x. This result matches the analytic solution above, but has a constant of integration `k = 2`.