积分学
积分
基本函数
\[
\int 0 \, \text{d}x = C
\]
\[
\int x^a \, \text{d}x = \frac{x^{a+1}}{a+1} + C \, (a \neq -1)
\]
\[
\int \frac{1}{x} \, \text{d}x = \ln |x| + C \, (x \neq 0)
\]
\[
\int a^x \, \text{d}x = \frac{a^x}{\ln a} + C \, (a > 0, a \neq 1)
\]
\[
\int e^x \, \text{d}x = e^x + C
\]
三角函数
\[
\int \cos x \, \text{d}x = \sin x + C
\]
\[
\int \sin x \, \text{d}x = -\cos x + C
\]
\[
\int \tan x \, \text{d}x = \ln |\cos x| + C
\]
\[
\int \cot x \, \text{d}x = \ln |\sin x| + C
\]
\[
\int \sec x \, \text{d}x = \ln |\sec x + \tan x| + C = \ln \left| \tan \frac{x}{2} + \frac{\pi}{4} \right| + C
\]
\[
\int \csc x \, \text{d}x = \ln |\csc x - \cot x| + C = \ln \left| \tan \frac{x}{2} + \frac{\pi}{4} \right| + C
\]
反三角函数
\[
\int \frac{1}{\sqrt{1 - x^2}} \, \text{d}x = \arcsin x + C \, (\text{或} \, \arccos x + C)
\]
\[
\int \frac{1}{1 + x^2} \, \text{d}x = \arctan x + C \, (\text{或} \, \text{arccot} x + C)
\]
其他
\[
\int \frac{1}{\cos^2 x} \, \text{d}x = \int \sec^2 x \, \text{d}x = \tan x + C
\]
\[
\int \frac{1}{\sin^2 x} \, \text{d}x = \int \csc^2 x \, \text{d}x = -\cot x + C
\]
\[
\int \frac{1}{a^2 - x^2} \, \text{d}x = \frac{1}{2a} \ln \left| \frac{a + x}{a - x} \right| + C
\]
\[
\int \frac{x}{\sqrt{a^2 - x^2}} \, \text{d}x = -\sqrt{a^2 - x^2} + C
\]
\[
\int \sqrt{a^2 - x^2} \, \text{d}x = \frac{x}{2} \sqrt{a^2 - x^2} + \frac{a^2}{2} \arctan \frac{x}{\sqrt{a^2 - x^2}} + C
\]
\[
\int \frac{\text{d}x}{\sqrt{x^2 + a^2}} = \ln |x + \sqrt{x^2 + a^2}| + C
\]
\[
\int e^x \sin x \, \text{d}x = \frac{1}{2} e^x (\sin x - \cos x) + C
\]
\[
\int e^x \cos x \, \text{d}x = \frac{1}{2} e^x (\sin x + \cos x) + C
\]