Toolkit 102Basic IntegralsDefinition (Signed Area)∫abf(x) dx\int_a^b f(x)\,dx∫abf(x)dxThe definite integral represents the signed area between the graph of f(x) and the x-axis.Basic Properties∫aaf(x) dx=0\int_a^a f(x)\,dx=0∫aaf(x)dx=0∫abf(x) dx=−∫baf(x) dx\int_a^b f(x)\,dx=-\int_b^a f(x)\,dx∫abf(x)dx=−∫baf(x)dx∫acf(x) dx=∫abf(x) dx+∫bcf(x) dx\int_a^c f(x)\,dx=\int_a^b f(x)\,dx+\int_b^c f(x)\,dx∫acf(x)dx=∫abf(x)dx+∫bcf(x)dx∫ab(f(x)±g(x)) dx=∫abf(x) dx±∫abg(x) dx\int_a^b(f(x)\pm g(x))\,dx=\int_a^b f(x)\,dx\pm\int_a^b g(x)\,dx∫ab(f(x)±g(x))dx=∫abf(x)dx±∫abg(x)dx∫abcf(x) dx=c∫abf(x) dx\int_a^b c f(x)\,dx=c\int_a^b f(x)\,dx∫abcf(x)dx=c∫abf(x)dxCommon Antiderivatives∫xn dx=xn+1n+1+C(n≠−1)\int x^n\,dx=\frac{x^{n+1}}{n+1}+C\qquad(n\ne-1)∫xndx=n+1xn+1+C(n=−1)∫1x dx=ln∣x∣+C\int\frac1x\,dx=\ln|x|+C∫x1dx=ln∣x∣+C∫ex dx=ex+C\int e^x\,dx=e^x+C∫exdx=ex+C∫ax dx=axlna+C(a>0,a≠1)\int a^x\,dx=\frac{a^x}{\ln a}+C\qquad(a>0,a\ne1)∫axdx=lnaax+C(a>0,a=1)∫sinx dx=−cosx+C\int\sin x\,dx=-\cos x+C∫sinxdx=−cosx+C∫cosx dx=sinx+C\int\cos x\,dx=\sin x+C∫cosxdx=sinx+C∫tanx dx=−ln∣cosx∣+C\int\tan x\,dx=-\ln|\cos x|+C∫tanxdx=−ln∣cosx∣+C∫cotx dx=ln∣sinx∣+C\int\cot x\,dx=\ln|\sin x|+C∫cotxdx=ln∣sinx∣+C
