\(\int x^{\alpha} \; \mathrm{d}x = \dfrac{x^{\alpha+1}}{\alpha+1}+C \space (\alpha \ne -1)\)

\(\int \dfrac{1}{x} \; \mathrm{d}x = \ln x+C \space\)

$\int {\alpha}^{x} \ \mathrm{d}x = \dfrac{{\alpha}^{x}}{\ln a}+C $ ，特别地 $\int {e}^{x} \ \mathrm{d}x = e^x +C $

\(\int \log_{a}x \; \mathrm{d}x = \dfrac{x\ln x-x}{\ln a}+C\) ，特别地 $\int \ln x \ \mathrm{d}x = x \ln x -x +C $

\(\int \sin x \; \mathrm{d}x = -\cos x+C \space\)

\(\int \cos x \; \mathrm{d}x = \sin x+C \space\)

\(\int \tan x \; \mathrm{d}x = \ln |\sec x|+C \space\)

\(\int \cot x \; \mathrm{d}x = -\ln |\csc x|+C \space\)

\(\int \sec x \; \mathrm{d}x = \ln|\sec x + \tan x| +C \space\)

\(\int \csc x \; \mathrm{d}x = \ln|\csc x - \cot x| +C \space\)

\(\int \sec x \tan x \; \mathrm{d}x = \sec x +C\)

\(\int \csc x \cot x \; \mathrm{d}x = -\csc x +C\)

\(\int \sin^2 x \ \mathrm{d}x = \dfrac{x}{2}-\dfrac{\sin {2x}}{4} +C\)

\(\int \tan^2 x \ \mathrm{d}x = \tan x -x +C\)

\(\int \sec^2 x \ \mathrm{d}x = \tan x +C\)

\(\int \sin^3 x \ \mathrm{d}x = \dfrac{\cos^3 x}{3} -\cos x +C\)

\(\int \tan^3 x \ \mathrm{d}x = \dfrac{\sec^2 x}{2} - \ln{|\sec x|} +C\)

\(\int \sec^3 x \ \mathrm{d}x = \dfrac{\sec x \tan x}{2} + \dfrac{\ln{|\sec x + \tan x|}}{2} +C\)

\(\int \arcsin x \ \mathrm{d}x = x \arcsin x + \sqrt{1-x^2} +C\)

\(\int \arccos x \ \mathrm{d}x = x \arccos x - \sqrt{1-x^2} +C\)

\(\int \arctan x \ \mathrm{d}x = x \arctan x - \dfrac{\ln(1+x^2)}{2} +C\)

\(\int \operatorname{arccot} x \ \mathrm{d}x = x \operatorname{arccot} x + \dfrac{\ln(1+x^2)}{2} +C\)

\(\int \operatorname{arcsec} x \ \mathrm{d}x = x \operatorname{arcsec} x - \ln(|x|+\sqrt{x^2-1}) +C\)

\(\int \operatorname{arccsc} x \ \mathrm{d}x = x \operatorname{arccsc} x + \ln(|x|+\sqrt{x^2-1}) +C\)

\(\int \dfrac{1}{\sqrt{x^2 \pm a^2}} \; \mathrm{d}x = \ln |x + \sqrt{x^2 \pm a^2}| +C \space\)

\(\int \dfrac{1}{\sqrt{a^2 - x^2}} \; \mathrm{d}x = \arcsin \dfrac{x}{a} +C \space\)

\(\int \dfrac{1}{x^2 + a^2} \; \mathrm{d}x = \dfrac{1}{a} \arcsin \dfrac{x}{a} +C \space\)

\(\int \dfrac{1}{x^2 - a^2} \; \mathrm{d}x = \dfrac{1}{2a} \ln |\dfrac{x-a}{x+a}| +C \space\)

\(\int \sqrt{x^2 \pm a^2} \; \mathrm{d}x = \dfrac{1}{2}( \; x \sqrt{x^2 \pm a^2} \pm a^2 \ln {|x + \sqrt{x^2 \pm a^2}|}) +C \space\)

\(\int \sqrt{a^2 - x^2} \; \mathrm{d}x = \dfrac{1}{2}( \; x \sqrt{a^2 - x^2} + a^2 \arcsin \dfrac{x}{a}) +C \space\)

部分题目求解过程：

• \(\int \tan x \ \mathrm{d}x = \int \dfrac{\sin x}{\cos x} \ \mathrm{d}x = \int \dfrac{-1}{\cos x} \ \mathrm{d}{(\cos x)} = - \ln |\cos x|+C = \ln |\sec x|+C\)

• $\int \sec x \ \mathrm{d}x = \int \dfrac{\sec x \ (\sec x + \tan x)}{\sec x + \tan x} \ \mathrm{d}x = \int \dfrac{\mathrm{d}{(\sec x + \tan x)} }{\sec x + \tan x} = \ln|\sec x + \tan x| +C $

• \(\int \sin^2 x \ \mathrm{d}x = \int \frac{1-\cos {2x}}{2} \ \mathrm{d}x = \frac{x}{2}-\frac{\sin {2x}}{4} +C\)

• \(\int \tan^2 x \ \mathrm{d}x = \int (\sec^2 x -1) \ \mathrm{d}x = \tan x -x +C\)

• \(\int \sin^3 x \ \mathrm{d}x = \int \sin x (1 - \cos^2 x) \ \mathrm{d}x = \int (\cos^2 x - 1) \ \mathrm{d}{(\cos x)} = \dfrac{\cos^3 x}{3} -\cos x +C\)

• \(\int \tan^3 x \ \mathrm{d}x = \int (\sec^2 x -1)(\tan x) \ \mathrm{d}x = \int \sec^2 x \tan x \ \mathrm{d}x - \int \tan x \ \mathrm{d}x = \int \sec x \ \mathrm{d}{(\sec x)}- \int \tan x \ \mathrm{d}x = \dfrac{\sec^2 x}{2} - \ln{|\sec x|} +C\)

• \(\int \sec^3 x \ \mathrm{d}x = \int \sec x \ \mathrm{d}{(\tan x)} = \sec x \tan x - \int \tan x \ \mathrm{d}{(\sec x)} = \sec x \tan x - \int (\sec^2 -1)\sec x \ \mathrm{d}{x} = \sec x \tan x - \int \sec^3 \ \mathrm{d}{x} + \int \sec x \ \mathrm{d}x \Rightarrow \int \sec^3 x \ \mathrm{d}x = \dfrac{\sec x \tan x}{2} + \dfrac{\ln{|\sec x + \tan x|}}{2} +C\)

• \(\int \arcsin x \ \mathrm{d}x = x \arcsin x - \int x \ \mathrm{d}{(\arcsin x)} = x \arcsin x + \cos{\arcsin x} +C = x \arcsin x + \sqrt{1 - x^2} +C\)

• \(\int \arctan x \ \mathrm{d}x = x \arctan x - \int x \ \mathrm{d}{(\arctan x)} = x \arctan x - \int \dfrac{x}{1+x^2} \ \mathrm{d}x = x \arctan x - \dfrac{1}{2} \int \dfrac{ \mathrm{d}(1+{x^2}) }{1+x^2} = x \arctan x - \dfrac{\ln(1+x^2)}{2} +C\)