常见函数积分

• $\int k\text{d}x=kx+C$
• $\int x^{\mu }\text{d}x=\frac{x^{\mu +1}}{\mu +1}+C(\mu \ne 1)$
• $\int \frac{\text{d}x}{x}=\text{ln}|x|+C$
• $\int \frac{\text{d}x}{1+x^2}=arctanx+C$
• $\int \frac{\text{d}x}{\sqrt{1-x^2}}=arcsinx+C$
• $\int cosx\text{d}x=sinx+C$
• $\int sinx\text{d}x=-cosx+C$
• $\int \frac{\text{d}x}{co{s}^{2}x}=tanx+C$
• $\int \frac{\text{d}x}{si{n}^{2}x}=-cotx+C$
• $\int secxtanx\text{d}x=secx+C$
• $\int cscxcotx\text{d}x=-cscx+C$
• $\int e^x\text{d}x=e^x+C$
• $\int a^x\text{d}x=\frac{a^x}{\text{ln}a}+C$
• $\int tanx\text{d}x=-\text{ln}|cosx|+C$
• $\int cotx\text{d}x=\text{ln}|sinx|+C$
• $\int \frac{\text{d}x}{a^2+x^2}=\frac{1}{a}arctan\frac{x}{a}+C$
• $\int \frac{\text{d}x}{\sqrt{a^2-x^2}}=arcsin\frac{x}{a}+C$
• $\int \frac{\text{d}x}{\sqrt{a^2+x^2}}=\text{ln}(x+\sqrt{a^2+x^2})+C$
• $\int \frac{\text{d}x}{\sqrt{x^2-a^2}}=\text{ln}|x+\sqrt{x^2-a^2}|+C$

第一类换元法：
找到 $f(x)=g[\phi (x)]{\phi }^{\prime }(x)$
$\int f(x)\text{d}x=\int g[\phi (x)]{\phi }^{\prime }(x)\text{d}x=\int g[\phi (x)]\text{d}\phi (x)=[\int g(u)\text{d}u{]}_{u=\phi (x)}$
第二类换元法：
找到 $x=\phi (t)$
$\int f(x)\text{d}x=[\int f(\phi (t)){\phi }^{\prime }(t)\text{d}t{]}_{t={\phi }^{-1}(x)}$