How to Find the Definite Integral of a Polynomial

Integrates a polynomial from a lower to an upper bound. Coefficients run from the highest power down, so 3, 2, 1 means 3x² + 2x + 1. The antiderivative of xⁿ is xⁿ⁺¹ ÷ (n+1), and the integral is the difference of its values at the two bounds.

This integrates a polynomial from a lower bound to an upper one. The antiderivative comes first, and then the difference of its values at the two bounds.

\int x^n dx = \dfrac{x^{n+1}}{n+1}

This is differentiation in reverse: raise the exponent by one and divide by that number. Enter the coefficients from the highest power down, so $3, 2, 1$ means $3x^2 + 2x + 1$ .

\int_a^b f(x)dx = F(b) - F(a)

$F$ is the antiderivative. This is the fundamental theorem of calculus.

Example

Integrate $3x^2 + 2x + 1$ from 0 to 2. The antiderivative is $x^3 + x^2 + x$ . At the upper bound it is $8 + 4 + 2 = 14$ and at the lower bound 0, so the integral is $14 - 0 = 14$ .

What the integral measures

Where $f(x)$ is positive, the integral is the area between the graph and the $x$ axis. Where $f(x)$ is negative, that area is added with a minus sign. When the graph crosses the axis, the positive and negative parts cancel each other out.

Notes

The constant of integration is taken as 0. Since the integral is a difference, any value would give the same answer.

An upper bound below the lower bound is allowed, and simply flips the sign.

Include a 0 for any missing power. For $x^3 + 1$ , enter $1, 0, 0, 1$ .