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Integrals formula sheet

All 19 Integrals formulas in CalcRef on one printable page, each with its conditions. Fundamental theorem, antiderivatives, and integral properties.

Integrals formula sheet

19 rows · CalcRef · sample of the first 6 rows

Integrals
FormulaStatementConditions
Riemann Sum∫abf(x) dx=lim⁡n→∞∑i=1nf(xi∗)Δx\int_a^b f(x)\, dx = \lim_{n \to \infty} \sum_{i=1}^{n} f(x_i^*) \Delta xno restriction stated
FTC Part 1ddx[∫axf(t) dt]=f(x)\frac{d}{dx}\left[\int_a^x f(t)\, dt\right] = f(x)f must be continuous on [a, x].
FTC Part 2∫abf(x) dx=F(b)−F(a) where F′(x)=f(x)\int_a^b f(x)\, dx = F(b) - F(a) \text{ where } F'(x) = f(x)f must be continuous on [a, b]. F is any antiderivative of f.
Linearity of Integrals∫ab[αf(x)+βg(x)] dx=α∫abf(x) dx+β∫abg(x) dx\int_a^b [\alpha f(x) + \beta g(x)]\, dx = \alpha \int_a^b f(x)\, dx + \beta \int_a^b g(x)\, dxno restriction stated
Additivity of Integrals∫abf(x) dx+∫bcf(x) dx=∫acf(x) dx\int_a^b f(x)\, dx + \int_b^c f(x)\, dx = \int_a^c f(x)\, dxno restriction stated
Reversal of Limits∫abf(x) dx=−∫baf(x) dx\int_a^b f(x)\, dx = -\int_b^a f(x)\, dxno restriction stated

The 19 Integrals formulas in the CalcRef library, every one of them free to read at /topics/integrals.

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