Mr. Integral
Math notes/Calculus I/The Integral

Antiderivatives and Initial Values — Cheat sheet

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Core rule

An antiderivative satisfies F′=fF'=f. All antiderivatives on one interval are F+CF+C.

∫xpdx=xp+1/(p+1)+C (p≠−1),\int x^pdx=x^{p+1}/(p+1)+C\ (p\ne-1), ∫dx/x=ln⁡∣x∣+C.\int dx/x=\ln|x|+C.

Watch for

Each integration introduces a constant. Apply initial conditions afterward and verify by differentiation. Disconnected intervals can have independent constants.