Termwise Antiderivatives of Polynomial Expressions

Why Integrate a Polynomial Term by Term?

A polynomial may contain several powers of its variable, but its terms can be handled separately. This works because the derivative of a sum is the sum of the derivatives. To find an antiderivative, we can reverse that process: apply the power rule to each term, then combine the results into one family.

This method is useful because it turns one expression into a short sequence of familiar power-rule steps. It also makes the result easy to check: differentiate the proposed antiderivative and confirm that each original term returns.

Polynomials and the Power Rule

A polynomial in \(x\) is a finite sum of terms whose powers of \(x\) are nonnegative integers. A general polynomial can be written as

where the coefficients \(a_0,a_1,\ldots,a_n\) are real numbers and \(n\) is a nonnegative integer. Some coefficients may be zero, so a polynomial does not have to contain every power between \(0\) and \(n\). A constant is also a polynomial: it is the case in which there are no positive powers of \(x\).

For a nonnegative integer \(k\), the power rule for antiderivatives says that

Here \(C\) is an arbitrary real constant. The rule applies because \(k+1\ne 0\). For a term with coefficient \(a\), the constant multiple rule gives

Thus, each term keeps its coefficient, its exponent increases by one, and its coefficient is divided by the new exponent.

The Termwise Rule

For a polynomial written in powers of \(x\), the termwise rule is

For the constant term \(a_0\), the power is \(x^0\). Applying the same rule gives \(a_0x^{0+1}/(0+1)=a_0x\). This is why a constant term becomes a linear term.

There is only one arbitrary constant in the final family. If each term is integrated with its own constant, their sum is still just one arbitrary constant: for example, \(C_1+C_2+C_3\) can be renamed \(C\). Keeping a single \(C\) makes the family clear without changing its meaning.

Why the Rule Works

The justification follows from the derivative power rule and the linearity of differentiation. For each term \(a_kx^k\), the power rule gives an antiderivative \(\frac{a_k}{k+1}x^{k+1}\). Its derivative is

Now add these antiderivatives for all the terms. Because the derivative of a finite sum is the sum of the derivatives, the derivative of the resulting expression is the sum of the original terms, which is \(P(x)\). Adding a constant does not change that derivative.

To check the formula directly, differentiate the sum:

So the displayed expression is an antiderivative. It also gives the complete family: any two antiderivatives of the same function on an interval differ by a constant. In this case, their difference has derivative zero, so it is constant; therefore no other forms need to be added to the family.

Worked Examples

A Reliable Termwise Process

When integrating a polynomial, keep the coefficient, exponent, and constant term visible at every step. The following sequence prevents most arithmetic errors:

Common Mistakes and Their Corrections

A frequent error is to increase the exponent but forget to divide by the new exponent. For example, an incorrect calculation might say

This is wrong because differentiating \(3x^3\) gives \(9x^2\), not \(3x^2\). The correction is to increase the power from \(2\) to \(3\) and divide the coefficient by \(3\):

Another error is to treat a constant as though its antiderivative were still constant. A constant \(a\) is \(ax^0\), so its antiderivative is \(ax\), since \(\frac{d}{dx}(ax)=a\). Also avoid writing a separate arbitrary constant for every term in the final result. If several constants appear during the calculation, their sum is one arbitrary constant, so the answer should end with a single \(+C\).

Why This Method Matters

Termwise integration is a direct use of the linearity of the derivative in reverse. It gives an efficient method for a broad class of expressions and a clear way to check the result. The essential verification is not that the final polynomial “looks right,” but that differentiating it recovers every coefficient and power of the original polynomial.

For polynomial expressions, the method always applies: there are finitely many terms, and each term has a nonnegative integer power. The exponent increases by one, the coefficient is divided by that new exponent, and the one arbitrary constant accounts for the full family of antiderivatives.