Antiderivatives Of Negative Integer Powers

Negative Powers and Antiderivatives

A negative power represents a reciprocal. For example, \(x^{-2}=1/x^2\) and \(x^{-3}=1/x^3\), provided \(x\ne 0\). These functions can be integrated with the power rule for antiderivatives: increase the exponent by one, then divide by the new exponent.

There is one important restriction. The rule applies to integer exponents at most \(-2\), but not to exponent \(-1\). For \(x^{-1}=1/x\), increasing the exponent by one would produce zero, and division by zero is not defined. The reciprocal function therefore requires separate treatment.

Why the Domain Matters

Negative powers are not defined at \(x=0\). For instance, \(x^{-2}=1/x^2\) would require division by zero there. Thus, an antiderivative of \(x^n\) for a negative integer exponent is considered on an interval that lies entirely to one side of zero, such as \((0,\infty)\) or \((-\infty,0)\), or on a smaller interval contained in one of these.

The interval condition is not just a technical detail. An antiderivative is a differentiable function on the domain under consideration, and the function \(x^n\) itself is undefined at zero. If a problem asks for an indefinite integral without specifying an interval, the formula is understood on any interval where the integrand is defined.

For negative exponents, it can help to translate between exponent form and fraction form. The translation does not change the integration rule:

After integrating, the result may also be left in exponent form or written as a reciprocal. For example, \(-x^{-1}\) and \(-1/x\) are the same expression wherever \(x\ne0\).

Deriving the Rule

The formula is a direct consequence of the differentiation power rule. For a differentiable power \(x^m\), its derivative is \(m x^{m-1}\), where the expression is defined. To find an antiderivative of \(x^n\), we want a power whose derivative has exponent \(n\). Raising the exponent by one gives \(x^{n+1}\), and differentiating this produces the factor \(n+1\).

This calculation explains both parts of the rule: increase the exponent by one, then divide by that new exponent. The condition \(n\le -2\) ensures that the new exponent \(n+1\) is not zero. The constant \(C\) is included because adding a constant does not change a derivative.

As a small illustration, take \(n=-3\). Raising the exponent by one gives \(-2\), and division by \(-2\) produces \(-x^{-2}/2\). Differentiating this result returns \(x^{-3}\), as the derivation predicts.

Worked Examples

The Exception: Exponent \(-1\)

The boundary of the rule is easy to miss. If \(n=-1\), then increasing the exponent by one gives \(0\), and the proposed denominator is also \(0\). Substituting \(n=-1\) into the formula is therefore invalid:

which is undefined. This does not mean that the reciprocal function has no antiderivative. It means only that the power rule in the displayed form does not provide it. The reciprocal function \(1/x\) and its logarithmic antiderivatives are treated separately.

Distinguishing the exponents is important when an integrand contains several terms. A term \(x^{-2}\) can be integrated with this rule, while a term \(x^{-1}\) cannot. Apply the rule only to the terms that meet its condition; do not force every negative exponent into one formula.

Common Pitfall: Losing a Negative Sign

A frequent error is to raise the exponent correctly but mishandle the division by the new, negative exponent. For example, a mistaken calculation for \(x^{-3}\) might be:

The new exponent is \(-2\), not \(2\). The denominator must retain its negative sign. The corrected calculation is:

Differentiate the incorrect answer to see the problem:

which has the wrong sign. Differentiate the corrected answer instead:

Checking by differentiation catches this kind of sign error immediately. It also verifies that each coefficient and exponent in a proposed antiderivative is correct.

A Reliable Procedure

When integrating a negative integer power, use this sequence of checks:

This procedure is particularly useful for sums. Handle each eligible power separately, preserve its coefficient and sign, and then differentiate the entire result as a final check.

Why This Rule Is Useful

Negative powers often appear when quantities are written as reciprocals or when algebraic expressions are simplified. Rewriting \(1/x^k\) as \(x^{-k}\) makes the power rule available for \(k\ge2\), since then the exponent \(-k\) is at most \(-2\). This gives a consistent way to integrate terms such as \(1/x^2\), \(1/x^3\), and \(1/x^4\).

The key is to use the rule only where its conditions hold. For \(x^n\) with integer \(n\le-2\), raising the exponent by one gives a nonzero denominator, and differentiation confirms the result. At exponent \(-1\), that denominator would be zero, so the reciprocal function must be handled by a different rule. Keeping this distinction clear prevents a plausible-looking but undefined calculation.