常用不定积分公式

\[ \begin{array}{ll} \displaystyle \int \sin x \mathrm{d}x = - \cos x + C & \displaystyle \int \cos x \mathrm{d}x = \sin x + C\\ \displaystyle \int \tan x \mathrm{d} x = - \ln |\cos x| + C& \displaystyle \int \cot x \mathrm{d}x = \ln |\sin x| + C \end{array} \]

\[ \begin{array}{ll} \displaystyle\int \sec x \mathrm{d} x = \ln|\sec x + \tan x| + C & \displaystyle \int \csc x \mathrm{d} x = - \ln |\csc x + \cot x| + C \\ \displaystyle \int \sec^2 x \mathrm{d}x = \tan x + C&\displaystyle \int \csc^2 x \mathrm{d}x = - \cot x + C\\ \end{array} \]

\[ \begin{array}{cc} \displaystyle \int \sec x \tan x \mathrm{d}x = \sec x + C&\displaystyle \int \csc x\cot x \mathrm{d}x = - \csc x + C\\ \end{array} \]

\[ \begin{array}{ll} \displaystyle\int \frac{1}{x^2 + a^2} \mathrm{d}x = \frac{1}{a} \arctan \frac{x}{a} + C &\displaystyle \int \frac{1}{x^2 - a^2}\mathrm{d}x = \frac{1}{2a} \ln |\frac{x-a}{x+a}| + C\\ \displaystyle\int \frac{1}{\sqrt{x^2 + a^2}} \mathrm{d}x = \ln |x + \sqrt{x^2 + a^2}|+C&\displaystyle \int \frac{1}{\sqrt{x^2 - a^2}} \mathrm{d}x = \ln |x + \sqrt{x^2 - a^2}| + C \\ \displaystyle\int \frac{1}{\sqrt{a^2 - x^2}}\mathrm{d}x = \arcsin \frac{x}{a} + C& \\ \end{array} \]

\[ \begin{array}{ll} \displaystyle \int \ln x \mathrm{d} x = x \ln x - x + C&\displaystyle \int \ln(1 + x)\mathrm{d} x = (1+x) \ln(1+x) - x + C \\ \end{array} \]

\[ \begin{array}{ll} \displaystyle \int \sqrt{x^2 \pm a^2}\mathrm{d} x = \frac{1}{2} \left[ x \sqrt{x^2 \pm a^2} \pm a^2 \ln|x + \sqrt{x^2 \pm a^2}| \right] + C\\ \displaystyle \int \sqrt{a^2 - x^2}\mathrm{d} x = \frac{1}{2} \left[ x \sqrt{a^2 - x^2} + a^2 \arcsin \frac{x}{a} \right] + C \end{array} \]