How Indefinite Integrals Work
An indefinite integral finds the antiderivative of a function — the function whose derivative gives back the original. Because differentiating kills constant terms, every antiderivative comes with an unknown constant + C. The power rule does most of the heavy lifting: ∫x^n dx = x^(n+1)/(n+1), for any n except −1.
The Core Rules
- Power rule: ∫x^n dx = x^(n+1)/(n+1) + C (n ≠ −1).
- Special case: ∫1/x dx = ln|x| + C.
- Exponentials: ∫e^x dx = e^x + C; ∫a^x dx = a^x/ln(a) + C.
- Trigonometry: ∫sin(x) dx = −cos(x) + C; ∫cos(x) dx = sin(x) + C.
- Constant multiple: ∫k·f(x) dx = k·∫f(x) dx — pull constants out first.
Beyond the Basic Table
Products, quotients, and compositions need substitution, parts, or partial fractions — techniques this table does not attempt. For a numeric answer over an interval, use our definite integral calculator instead. To check an answer, differentiate it with our derivative calculator — you should get the integrand back.