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Cyletix大约 1 分钟

幂指函数

∫kdx=kx+C \int k dx = kx + C

\int x^n dx = \frac{x^{n+1}}{n+1} + C \quad (n \neq -1)$$$$\int \frac{1}{x} dx = \ln|x| + C

∫exdx=ex+C \int e^x dx = e^x + C

∫axdx=axln⁡a+C(a>0,a≠1) \int a^x dx = \frac{a^x}{\ln a} + C \quad (a > 0, a \neq 1)

三角函数

∫sin⁡xdx=−cos⁡x+C \int \sin x dx = -\cos x + C

∫cos⁡xdx=sin⁡x+C \int \cos x dx = \sin x + C

∫tan⁡xdx=−ln⁡∣cos⁡x∣+C \int \tan x dx = -\ln|\cos x| + C

∫cot⁡xdx=ln⁡∣sin⁡x∣+C \int \cot x dx = \ln|\sin x| + C

∫sec⁡xdx=ln⁡∣sec⁡x+tan⁡x∣+C \int \sec x dx = \ln|\sec x + \tan x| + C

∫csc⁡xdx=ln⁡∣csc⁡x−cot⁡x∣+C \int \csc x dx = \ln|\csc x - \cot x| + C

∫sec⁡2xdx=tan⁡x+C \int \sec^2 x dx = \tan x + C

∫csc⁡2xdx=−cot⁡x+C \int \csc^2 x dx = -\cot x + C

∫1sin⁡2xdx=−cot⁡x+C \int \frac{1}{\sin^2 x} dx = -\cot x + C

∫11−sin⁡xdx=−ln⁡∣1−sin⁡x−cos⁡x∣+C \int \frac{1}{1 - \sin x} dx = -\ln|1 - \sin x - \cos x| + C

反三角函数

∫11−x2dx=arcsin⁡x+C \int \frac{1}{\sqrt{1-x^2}} dx = \arcsin x + C

∫1a2−x2dx=arcsin⁡xa+C(a>0) \int \frac{1}{\sqrt{a^2 - x^2}} dx = \arcsin \frac{x}{a} + C \quad (a > 0)

∫−11−x2dx=arccos⁡x+C \int -\frac{1}{\sqrt{1-x^2}} dx = \arccos x + C

∫11+x2dx=arctan⁡x+C \int \frac{1}{1+x^2} dx = \arctan x + C

∫−1x2+1dx=arccot x+C \int -\frac{1}{x^2+1} dx = \text{arccot} \, x + C

∫1xx2−1dx=arcsec x+C \int \frac{1}{x\sqrt{x^2-1}} dx = \text{arcsec} \, x + C

∫−1xx2−1dx=arccsc x+C \int -\frac{1}{x\sqrt{x^2-1}} dx = \text{arccsc} \, x + C

反双曲函数

∫1x2+a2 dx=ln⁡∣x+x2+a2∣+C=arcsinh(xa)+C \int \frac{1}{\sqrt{x^2 + a^2}} \, dx = \ln|x + \sqrt{x^2 + a^2}| + C = \text{arcsinh}\left(\frac{x}{a}\right) + C

∫1x2−a2 dx=ln⁡∣x+x2−a2∣+C=arccosh(xa)+C \int \frac{1}{\sqrt{x^2 - a^2}} \, dx = \ln|x + \sqrt{x^2 - a^2}| + C = \text{arccosh}\left(\frac{x}{a}\right) + C

∫11−x2 dx=12ln⁡∣1+x1−x∣+C=arctanh(x)+C(∣x∣<1) \int \frac{1}{1 - x^2} \, dx = \frac{1}{2} \ln\left|\frac{1 + x}{1 - x}\right| + C = \text{arctanh}(x) + C \quad (|x| < 1)

∫1x2−1 dx=12ln⁡∣x−1x+1∣+C=arccoth(x)+C(∣x∣>1) \int \frac{1}{x^2 - 1} \, dx = \frac{1}{2} \ln\left|\frac{x - 1}{x + 1}\right| + C = \text{arccoth}(x) + C \quad (|x| > 1)

∫1x1−x2 dx=−ln⁡∣x+x2−1∣+C=arcsech(x)+C(0<x<1) \int \frac{1}{x \sqrt{1 - x^2}} \, dx = -\ln\left|x + \sqrt{x^2 - 1}\right| + C = \text{arcsech}(x) + C \quad (0 < x < 1)

∫1∣x∣1+x2 dx=ln⁡∣x+x2+1∣+C=arccsch(x)+C \int \frac{1}{|x| \sqrt{1 + x^2}} \, dx = \ln\left|x + \sqrt{x^2 + 1}\right| + C = \text{arccsch}(x) + C

反圆锥曲线函数与对数函数的统一 反圆锥曲线函数与对数函数的统一