Antiderivatives and the indefinite integral

The Fundamental Theorem says you evaluate integrals by finding an antiderivative — a function whose derivative is the one you're integrating. So integration, in practice, is differentiation run in reverse.

Reversing the power rule

To differentiate a power you multiply by the exponent and drop it by one. To antidifferentiate, do the opposite — raise the exponent by one and divide by the new exponent:

∫ xⁿ dx = xⁿ⁺¹ / (n + 1) + C (for n ≠ −1)

Check it by differentiating the answer: d/dx [ xⁿ⁺¹/(n+1) ] = (n+1)xⁿ/(n+1) = xⁿ. ✓

Where the "+ C" comes from

Here's the catch that indefinite integrals must carry. The derivative of a constant is zero, so x² + 1, x² + 5, and x² − 3 all have the same derivative, 2x. Reversing 2x can't tell which constant was there — so we write the whole family at once with an arbitrary + C:

∫ 2x dx = x² + C.

Every curve in that family is a vertical shift of the others: same slope everywhere, same derivative.

A small table

The 1/x case is the exception the power rule can't touch (it would divide by zero) — its antiderivative is the natural log.

Why this matters

Antiderivatives are the engine behind the Fundamental Theorem: every definite integral you evaluate is "find the antiderivative, plug in the endpoints, subtract." Getting fluent at reversing derivatives is what makes integration practical — and it sets up the techniques in the next module.