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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,

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

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.