Basic Integral Formulas

Power functions

Formula: $\int x^n dx = \dfrac{x^{n+1}}{n+1} + C$ ($n\ne -1$)
Special: $\int \dfrac{1}{x} dx = \ln|x| + C$.

Exponential

• $\int e^x dx = e^x + C$
• $\int a^x dx = \dfrac{a^x}{\ln a} + C$ ($a>0$, $a\ne1$)

Examples: $\int e^{2x} dx = \tfrac12 e^{2x} + C$, $\int 2^x dx = \dfrac{2^x}{\ln 2}+C$.

Logarithmic

$\int \dfrac{1}{x} dx = \ln|x| + C$ (absolute value for domain).

Trigonometric

• $\int \sin x dx = -\cos x + C$
• $\int \cos x dx = \sin x + C$
• $\int \tan x dx = -\ln|\cos x| + C$
• $\int \cot x dx = \ln|\sin x| + C$
• $\int \sec x dx = \ln|\sec x + \tan x| + C$
• $\int \csc x dx = -\ln|\csc x + \cot x| + C$

Inverse trig

• $\int \dfrac{1}{\sqrt{1-x^2}} dx = \arcsin x + C$
• $\int \dfrac{1}{1+x^2} dx = \arctan x + C$
• $\int \dfrac{1}{|x|\sqrt{x^2-1}} dx = \arccos x + C$ ($|x|>1$)

Hyperbolic

• $\int \sinh x dx = \cosh x + C$
• $\int \cosh x dx = \sinh x + C$
• $\int \tanh x dx = \ln|\cosh x| + C$

Memory tips

• Integrals are inverse derivatives—verify by differentiating.
• Group by type: power, exponential, log, trig, inverse trig, hyperbolic.

练习题

练习 1

计算 $\int x^4 dx$。

练习 2

计算 $\int \dfrac{1}{x^3} dx$。

练习 3

计算 $\int e^{5x} dx$。

练习 4

计算 $\int \cos x dx$。

Summary

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中英对照