Integral of 1/x²: Formula, Derivation and Examples

The integral of \dfrac{1}{x^2} is -\dfrac{1}{x} + C. Unlike \frac{1}{x}, this one is handled by the ordinary reverse power rule.

Graph of 1/x² and the area under it

Derivation, step by step

1. Rewrite as a power: \frac{1}{x^2} = x^{-2}.
2. Apply the reverse power rule with n = -2.
3. Add one to the exponent: -2 + 1 = -1.
4. Divide by -1: \frac{x^{-1}}{-1} = -\frac{1}{x}.
5. Add C.

Check by differentiating

The forward rule is on the derivative of 1/x² page, and the derivative of 1/x page shows that \frac{d}{dx}\frac{1}{x} = -\frac{1}{x^2}, which is the same fact read the other way.

Improper integrals

The area under \frac{1}{x^2} out to infinity is finite:

Compare this with \int_1^\infty \frac{1}{x}\,dx, which diverges because \ln b \to \infty. This contrast is the start of the p-test for integrals and series. The lesson on infinite limits of integration covers it in detail.

Near zero the story flips: \int_0^1 \frac{1}{x^2}\,dx = \lim_{a\to 0^+}\left(\frac{1}{a} - 1\right) = \infty.

The classic trap: integrating across zero

Plugging into the antiderivative gives

But \frac{1}{x^2} is positive everywhere, so its integral cannot be negative. The error is that \frac{1}{x^2} has a vertical asymptote at x = 0, so the Fundamental Theorem of Calculus does not apply on [-1, 1]. Split the integral at 0: each half diverges, so the whole integral diverges.

Substitution variant

With u = ax + b:

For example, \int \frac{1}{(2x + 1)^2}\,dx = -\frac{1}{2(2x + 1)} + C. Check: the derivative is \frac{1}{2}(2x + 1)^{-2}\cdot 2 = \frac{1}{(2x + 1)^2}.

Worked examples

Example 1. \int \frac{3}{x^2}\,dx = -\frac{3}{x} + C.

Example 2. \int \frac{x + 1}{x^2}\,dx = \int\left(\frac{1}{x} + \frac{1}{x^2}\right)dx = \ln\lvert x\rvert - \frac{1}{x} + C. Split first, then use the integral of 1/x for the first piece.

Example 3. \int_2^4 \frac{1}{x^2}\,dx = -\frac{1}{4} + \frac{1}{2} = \frac{1}{4}.

Example 4. \int \frac{1}{x^3}\,dx = \frac{x^{-2}}{-2} + C = -\frac{1}{2x^2} + C.

Common mistakes

1. Writing \ln(x^2). The log only appears for \frac{1}{x}.
2. Writing -\frac{1}{x^3} or \frac{1}{x}. Exponents go up by one (-2 \to -1), and you divide by -1, which flips the sign.
3. Integrating across an asymptote. Always check for division by zero inside the interval.
4. Writing \frac{1}{3x^3} by treating \frac{1}{x^2} as (x^2) and integrating that. Rewrite as x^{-2} first.

Try improper integral problems to practise the infinite case.