Al.4 一些有用的标准积分

$f(x), (a > 0)$

$$\frac{1}{a^2 + x^2}$$

$$\frac{1}{\sqrt{a^2 - x^2}}$$

$$\frac{1}{\sqrt{a^2 + x^2}}$$

$$\frac{1}{\sqrt{x^2 - a^2}}$$

$$\frac{1}{a^2 - x^2} \quad \text{for } |x| < a$$

$$\frac{1}{x^2 - a^2} \quad \text{for } |x| > a$$

$f(x) dx$ $$\frac{1}{a} \tan^{-1}\left(\frac{x}{a}\right)$$

$$\sin^{-1}\left(\frac{x}{a}\right)$$

$$\sinh^{-1}\left(\frac{x}{a}\right) \quad \text{or} \quad \ln[x + \sqrt{x^2 + a^2}]$$

$$\cosh^{-1}\left(\frac{x}{a}\right) \quad \text{or} \quad \ln[x + \sqrt{x^2 - a^2}]$$

$$\frac{1}{2a} \ln\left(\frac{a + x}{a - x}\right)$$

$$\frac{1}{2a} \ln\left(\frac{x - a}{x + a}\right)$$