Antiderivatives

An antiderivative of f is a function whose derivative is f. Finding one means asking what would have differentiated to this.

The reversed Power Rule

Add one to the exponent and divide by the new exponent. The integral of x³ is x⁴ ÷ 4.

Always checkable

Differentiate your answer. If it returns the original function, the antiderivative is right.

The constant

Constants vanish when differentiated, so every antiderivative comes with a + C. There is a whole family of them.

A family of curves

Those antiderivatives are vertical shifts of one another. They all share the same slope at every x.

One exception

The rule fails for x⁻¹, since adding one gives a zero exponent and a division by zero. That case needs the logarithm.

Differentiation, run backwards

An antiderivative of f is a function whose derivative is f. Since the derivative of x² is 2x, an antiderivative of 2x is x². Every differentiation rule read backwards becomes an integration rule.

The constant of integration

x², x² + 1 and x² − 7 all have derivative 2x, so the antiderivative is x² + C. Constants vanish under differentiation, so they cannot be recovered. Omitting the C is not a minor slip — it discards infinitely many solutions.

The basic rules

The power rule reverses to xⁿ⁺¹/(n + 1), except for n = −1, where the answer is ln|x|. That exception exists because dividing by n + 1 would divide by zero, and it is why the logarithm appears in integration at all.

Integration is harder than differentiation

Every elementary function can be differentiated by rules; many cannot be integrated in closed form at all. Differentiation is mechanical, integration is a search — which is why integration tables and numerical methods exist.