Home/Calculus II/Definite integral and Fundamental Theorem of Calculus/PropertiesSection 3.2PropertiesFormulaCopy LaTeX∫aaf(x) dx=0\int_a^a f(x)\,dx=0∫aaf(x)dx=0View detail →FormulaCopy LaTeX∫abf(x) dx=−∫baf(x) dx\int_a^b f(x)\,dx=-\int_b^a f(x)\,dx∫abf(x)dx=−∫baf(x)dxView detail →FormulaCopy LaTeX∫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)\,dx∫abf(x)dx+∫bcf(x)dx=∫acf(x)dxView detail →FormulaCopy LaTeX∫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)\,dx∫ab[αf(x)+βg(x)]dx=α∫abf(x)dx+β∫abg(x)dxView detail →If f(x)≥0f(x)\ge0f(x)≥0 on [a,b][a,b][a,b], thenFormulaCopy LaTeX∫abf(x) dx≥0.\int_a^b f(x)\,dx\ge0.∫abf(x)dx≥0.View detail →If f(x)≤g(x)f(x)\le g(x)f(x)≤g(x) on [a,b][a,b][a,b], thenFormulaCopy LaTeX∫abf(x) dx≤∫abg(x) dx.\int_a^b f(x)\,dx\le\int_a^b g(x)\,dx.∫abf(x)dx≤∫abg(x)dx.View detail →
FormulaCopy LaTeX∫abf(x) dx=−∫baf(x) dx\int_a^b f(x)\,dx=-\int_b^a f(x)\,dx∫abf(x)dx=−∫baf(x)dxView detail →
FormulaCopy LaTeX∫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)\,dx∫abf(x)dx+∫bcf(x)dx=∫acf(x)dxView detail →
FormulaCopy LaTeX∫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)\,dx∫ab[αf(x)+βg(x)]dx=α∫abf(x)dx+β∫abg(x)dxView detail →
FormulaCopy LaTeX∫abf(x) dx≤∫abg(x) dx.\int_a^b f(x)\,dx\le\int_a^b g(x)\,dx.∫abf(x)dx≤∫abg(x)dx.View detail →