This example shows how to use the `polyint`

function to integrate polynomial expressions analytically. Use this function to evaluate indefinite integral expressions of polynomials.

Consider the real-valued indefinite integral,

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

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.

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 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`

.