To integrate a polynomial, you apply the power rule to each term individually, then combine the results. The power rule states that the integral of x raised to the power n equals x raised to (n+1), divided by (n+1), plus a constant. Once you know this single rule, you can integrate any polynomial, no matter how many terms it has.
The Power Rule
The core formula looks like this:
∫ xⁿ dx = xⁿ⁺¹ / (n+1) + C
In plain terms: raise the exponent by one, then divide by the new exponent. C is a constant you add at the end (more on that below). This works for any value of n, whether it’s positive, negative, or a fraction, with one exception: n cannot equal -1, because that would mean dividing by zero. If you ever encounter x⁻¹ (which is 1/x), the integral is ln|x| + C, not the power rule.
Integrating Term by Term
A polynomial is just a sum of terms, and the integral of a sum equals the sum of the integrals. That means you can handle each term on its own, then put everything back together. Here’s the process:
- Pull out any constant coefficient. If a term is 5x³, the 5 can move outside the integral. You integrate x³ and then multiply the result by 5.
- Apply the power rule to the variable part. Increase the exponent by 1 and divide by that new exponent.
- Repeat for every term in the polynomial.
- Add a single + C at the end. Even though each term technically generates its own constant, the sum of several unknown constants is still just one unknown constant.
A Worked Example
Suppose you need to integrate 3x⁴ + 2x² − 7x + 5. Take it one term at a time:
∫ 3x⁴ dx = 3 · x⁵/5 = 3x⁵/5
∫ 2x² dx = 2 · x³/3 = 2x³/3
∫ −7x dx = −7 · x²/2 = −7x²/2
∫ 5 dx = 5x
That last term trips people up. A plain constant like 5 is really 5x⁰. Applying the power rule gives 5x¹/1, which is simply 5x.
Combining everything: 3x⁵/5 + 2x³/3 − 7x²/2 + 5x + C.
Why You Need the + C
When you differentiate a constant, it vanishes. That means many different functions can share the same derivative. For instance, x² + 3 and x² + 100 both have the derivative 2x. Integration reverses differentiation, so when you compute ∫ 2x dx, you can’t know which original constant was there. Writing + C covers all possibilities at once.
This matters more than it might seem. In later coursework, the specific value of C gets pinned down by an initial condition or boundary value. If you’re told, for example, that the function equals 10 when x = 0, you plug in those values to solve for C. Forgetting it means losing information you’ll need later.
Definite Integrals: Evaluating Between Limits
Sometimes you’re given upper and lower bounds on the integral, written as ∫ from a to b. In that case you don’t need + C. Instead, you find the antiderivative (the same power rule result), plug in the upper limit, plug in the lower limit, and subtract.
The formula is F(b) − F(a), where F is the antiderivative. For example, to evaluate ∫ from 1 to 3 of 2x dx:
The antiderivative is x². Plug in the upper limit: 3² = 9. Plug in the lower limit: 1² = 1. The result is 9 − 1 = 8.
Be careful with the subtraction step. Always go upper limit minus lower limit, and use parentheses when the antiderivative has multiple terms. A misplaced negative sign is one of the most common errors in definite integral problems.
Common Mistakes to Avoid
A few errors come up again and again when integrating polynomials:
Forgetting the n = −1 restriction. If you blindly apply the power rule to x⁻¹, you get x⁰/0, which is undefined. The integral of 1/x is ln|x| + C, not the power rule. This won’t come up with a standard polynomial (which has only non-negative integer exponents), but it will appear the moment you work with expressions that have variables in the denominator.
Dividing by the old exponent instead of the new one. After integrating x³, the new exponent is 4, so you divide by 4, not 3. A quick check: differentiate your answer. If you get the original term back, you did it correctly.
Trying to split products the same way you split sums. You can integrate a sum term by term, but you cannot multiply two separate integrals together and expect to get the integral of a product. If two polynomial factors are multiplied, expand them first into a single polynomial, then integrate each resulting term.
Dropping the constant of integration. It feels like a minor bookkeeping detail, but leaving off + C means your answer represents only one of infinitely many valid antiderivatives.
Where Polynomial Integration Shows Up
If you’re learning this in a math class, it might feel purely abstract, but polynomial integrals appear constantly in applied fields. In physics, the basic motion equations for constant acceleration come from integrating polynomial expressions. Integrating a constant acceleration gives a velocity function (a linear polynomial), and integrating that gives a position function (a quadratic polynomial). The potential energy stored in a stretched spring, the moment of inertia of a rod or cylinder, and the energy stored in a capacitor all involve evaluating definite integrals of polynomial terms. In each case, the math is exactly what you’ve seen here: apply the power rule term by term, evaluate at the limits, and subtract.