# Documentation

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## 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,

 

The integrand is a polynomial, and the analytic solution is

 

where 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 = 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.