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Basic Integrals

Lesson · Beginner

Calculus

Definition (Signed Area)

Signed area represented by a definite integral
∫abf(x) dx\int_a^b f(x)\,dx

The 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
∫abf(x) dx=−∫baf(x) dx\int_a^b f(x)\,dx=-\int_b^a f(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
∫ab(f(x)±g(x)) dx=∫abf(x) dx±∫abg(x) dx\int_a^b\bigl(f(x)\pm g(x)\bigr)\,dx=\int_a^b f(x)\,dx\pm\int_a^b g(x)\,dx
∫abcf(x) dx=c∫abf(x) dx\int_a^b c f(x)\,dx=c\int_a^b f(x)\,dx

Common Antiderivatives

∫xn dx=xn+1n+1+C(n≠−1)\int x^n\,dx=\frac{x^{n+1}}{n+1}+C\qquad(n\ne-1)
∫1x dx=ln⁡∣x∣+C\int\frac1x\,dx=\ln|x|+C
∫ex dx=ex+C\int e^x\,dx=e^x+C
∫ax dx=axln⁡a+C(a>0, a≠1)\int a^x\,dx=\frac{a^x}{\ln a}+C\qquad(a>0,\ a\ne1)
∫sin⁡x dx=−cos⁡x+C\int\sin x\,dx=-\cos x+C
∫cos⁡x dx=sin⁡x+C\int\cos x\,dx=\sin x+C
∫tan⁡x dx=−ln⁡∣cos⁡x∣+C\int\tan x\,dx=-\ln|\cos x|+C
∫cot⁡x dx=ln⁡∣sin⁡x∣+C\int\cot x\,dx=\ln|\sin x|+C
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