积分的幂规则

For powers, there is a special rule regarding antidifferentiation:

$\int{x^ndx} = \frac{1}{n + 1}x^{n + 1} + C$

This can be checked with differentiation of the equation--if you have made it this far then the result should look familiar:

$x^n = \frac{d}{dx}\left(\frac{1}{n + 1}x^{n + 1} + C\right)$

Or, you may see it in this form, where both sides of the equation are multiplied by $n + 1$:

$(n + 1)x^n = \frac{d}{dx}(x^{n + 1} + C)$

Consider what happens in the original formula if we try to integrate $x^{-1}$:

$\int{x^{-1}dx} = \frac{1}{-1 + 1}x^{-1 + 1} + C = \frac{1}{0}x^0 + C$

We cannot divide by $0$, so this formula does not work for $x = -1$. Instead, this expression has a special antiderivative:

$\int{x^{-1}dx} = \ln|x| + C$

which in turn means

$\frac{d}{dx}(\ln(x)) = \frac{1}{x} = x^{-1}$