# Common Mistakes

\[
-x^{-2} &\neq \int x^{-1} = \int \frac{1}{x} = \ln x \\ 
\frac{1}{x} &\neq \int \ln x = x\ln x - x \\
\int x^{-k} = \frac{1}{-k+1}x^{-k+1} &\neq \frac{1}{-(k+1)}x^{-(k+1)} \\
\text{ e.g. } \int x^{-2} = -x^{-1} &\neq -\frac{1}{3}x^{-3}
\lim_{n\to\infty} \frac{n}{n+1} = 1 \neq 0\\
\frac{\partial}{\partial x}a^x = \frac{\partial}{\partial x}e^{x\ln a} = e^{x\ln a} \ln a = a^x \ln a.
\]

Exponentials: when in doubt, write $a^b = e^{b\ln a}$

\[
\frac{\partial}{\partial x} x^{f(x)} = ?
\]

\[
\sum x^k = \frac{1}{1-x} \neq \frac{1}{1+x} = \sum (-1)^k x^k
\]