不定积分公式大全24个
时间: 2023-08-21 16:03:11 浏览: 95
好的,以下是不定积分公式的大全:
1. 基本初等函数不定积分公式:
- $\int x^{\alpha}dx = \frac{x^{\alpha+1}}{\alpha+1} + C$,其中$\alpha\neq-1$
- $\int \frac{1}{x}dx = \ln|x| + C$
- $\int e^x dx = e^x + C$
- $\int \sin x dx = -\cos x + C$
- $\int \cos x dx = \sin x + C$
- $\int \sec^2 x dx = \tan x + C$
- $\int \csc^2 x dx = -\cot x + C$
- $\int \sec x\tan x dx = \sec x + C$
- $\int \csc x\cot x dx = -\csc x + C$
- $\int \frac{1}{1+x^2}dx = \arctan x + C$
- $\int \frac{1}{\sqrt{1-x^2}}dx = \arcsin x + C$
2. 常见函数的不定积分公式:
- $\int \frac{1}{a^2+x^2}dx = \frac{1}{a}\arctan\frac{x}{a} + C$
- $\int \frac{1}{\sqrt{a^2-x^2}}dx = \arcsin\frac{x}{a} + C$
- $\int \frac{1}{x\sqrt{x^2-a^2}}dx = \frac{1}{a}\ln\left|\frac{\sqrt{x^2-a^2}+x}{|a|}\right| + C$
- $\int \frac{1}{x\sqrt{x^2+a^2}}dx = \frac{1}{a}\ln\left|\frac{\sqrt{x^2+a^2}+a}{|x|}\right| + C$
- $\int \frac{1}{\sqrt{x^2\pm a^2}}dx = \ln|x+\sqrt{x^2\pm a^2}| + C$
- $\int \frac{1}{\sqrt{a^2-x^2}}dx = \arcsin\frac{x}{a} + C$
- $\int \frac{1}{\sqrt{x^2-a^2}}dx = \ln|x+\sqrt{x^2-a^2}| + C$
- $\int \frac{1}{a^2-x^2}dx = \frac{1}{2a}\ln\left|\frac{a+x}{a-x}\right| + C$
- $\int \frac{1}{x^2-a^2}dx = \frac{1}{2a}\ln\left|\frac{x-a}{x+a}\right| + C$
- $\int \frac{1}{(x-a)(x-b)}dx = \frac{1}{a-b}\ln\left|\frac{x-a}{x-b}\right| + C$
- $\int \frac{1}{(x-a)^n}dx = \frac{1}{(n-1)(x-a)^{n-1}} + C$,其中$n\neq1$
- $\int e^{ax}\sin bx dx = \frac{1}{a^2+b^2}e^{ax}(a\sin bx - b\cos bx) + C$
- $\int e^{ax}\cos bx dx = \frac{1}{a^2+b^2}e^{ax}(a\cos bx + b\sin bx) + C$
3. 特殊函数的不定积分公式:
- $\int \frac{1}{\sqrt{x^2\pm 1}}dx = \ln|x+\sqrt{x^2\pm 1}| + C$
- $\int \frac{1}{\sqrt{1-x^2}}\ln\left|\frac{1+\sqrt{1-x^2}}{x}\right|dx = -\frac{\pi}{2}\ln 2 + \frac{1}{2}\ln^2 2 + \arcsin x\ln\left|\frac{1+\sqrt{1-x^2}}{x}\right| + C$
- $\int \frac{1}{\sqrt{1-x^4}}dx = \frac{1}{2}\sqrt{2}\ln\left|\frac{\sqrt{2}x^2+1}{\sqrt{2}x^2-1}\right| + \frac{1}{\sqrt{2}}\arctan\left(\sqrt{2}x\sqrt{\frac{1+x^4}{2}}\right) + C$
- $\int \frac{1}{\sqrt{x^4\pm 1}}dx = \frac{1}{2}\ln\left|\frac{x^2+\sqrt{x^4\pm 1}}{x^2-\sqrt{x^4\pm 1}}\right| + C$
- $\int \sqrt{x^2\pm a^2}dx = \frac{1}{2}\left(x\sqrt{x^2\pm a^2}\pm a^2\ln|x+\sqrt{x^2\pm a^2}|\right) + C$
- $\int \frac{1}{\sqrt{x^2-a^2}\sqrt{x^2-b^2}}dx = \frac{1}{ab}\ln\left|\frac{\sqrt{x^2-a^2}-\sqrt{x^2-b^2}}{\sqrt{x^2-a^2}+\sqrt{x^2-b^2}}\right| + C$
希望这些公式能对你有所帮助!