💯
University Notes
  • Introduction
  • High Level Math
    • Function, limitation, and continuity
      • What is function?
      • Two kinds of infinity
      • The limitation of a function
      • A model for getting limitation
    • Derivative and differential
      • Formulas of derivative
      • Use derivative formula
      • Goes deeper
      • Use derivative
        • Function analyzing in theory
          • First derivative
          • Second derivative
          • Domain and Extreme Value
          • Overall change
        • Derivative use in reality
    • Integration
      • Indefinite integral
        • Basic formulas
        • Use formula
        • Goes deeper
        • Integration by parts
      • Definite integral
        • Properties of definite integral
        • Second fundamental theorem of calculus
        • Multi-method for solving definite integral
    • Multivariable calculus
      • Limitation
      • Partial derivatives
      • Differential
      • Multiple integral
    • Series
    • Linear algebra
    • GaoKao
      • 1
      • The road for starting
      • Polar Coordinates
      • Tangent Line
  • Electrical Engineering
    • The Terminologys
    • DC
      • The circuit rule
      • KCL and KVL
      • Superposition
    • AC
      • Intuition
      • Resistor
      • Inductor
      • Capacitor
      • AC circuit
      • 三相电
        • 星形联结
        • 三角形联结
        • 实际电路
    • Voltage and Current Rule in Circuit
    • Response
      • Foundations
      • 零输入响应
      • 零状态响应
      • 一阶电路的全响应
  • Analog Electronics
    • Technical terms 1
    • DC stable source circuit 的分析与应用
      • 二极管的特性与应用
        • 半导体
        • PN junction
        • Diode
        • 测试二极管
      • 整流滤波电路的分析与应用
        • Rectifier circuit
      • 直流稳压电路的分析
        • Zener diode
        • Shunt voltage regulators
    • Thyristor
    • Technical terms 2
    • Amplifying circuit
      • Bipolar Junction Transistor
      • Common Emitter Configuration
      • Biasing
      • Analysis
      • Mess
      • Negative-feedback amplifier
      • Integrated Operational Amplifier
    • Algorithms
      • What's the ouput of a voltage rectifier circuit
      • PNP or NPN
      • Judging the state of a BJT
      • What's common in BJT
      • Does a amplifying circuit normal
      • What's the feedback type
      • What kind of distortion you are encounter
  • Digital Electronic Technology
    • Logic Gate
    • Logic expressions
    • Karnaugh map
    • Number system
    • Multiplexer
    • Flip-flop
  • Principles of Communications
    • Overviews
    • PCM
    • HDB3
    • Modulations
    • Cyclic code
  • Data Communications and Networking
    • Something about IPv4
  • Micro Control System 51 Series
    • The Delay function
    • The Interrupt function
  • Maintenance of Railway Optical Cable Lines
    • Questions
    • Pictures
  • Mobile Communications
    • Concepts
    • Coding and Modulation
    • Key Technologies
    • Mobile communication network structure
    • Radio wave Propagation and Interference
    • GSM
    • CDMA
    • GPRS
    • 3G
    • 4G
    • Base Station Maintenance
  • Multimedia Communication
    • Concept of Multimedia
    • Compression
    • Lossless Compression
    • Audio
    • Lossy Audio Coding
    • Graph Compression
    • All for the exam
  • Power system for Communication Devices
    • Overview
    • AC power Distribution Panel
    • UPS
    • HF Switched-mode Power Supply
    • Battery
    • Earthing or Use Lightning Arrester
    • Power Supply Monitoring System
    • All for the exam
  • Optical fiber Communication system
    • What is Optical fiber Communication system
      • Prepare
      • Something About Optical fiber
      • Passive Optical Devices
      • Active Optical Devices
      • Optical transmitter Test
      • Optical receiver Test
      • Compose an Optical Communication System
    • SDH (Synchronous Digital Hierarchy)
      • Frame Structure of SDH
      • SDH Equipments
      • Clock System
      • ZXONM E300 Practice
      • SDH protection
    • WDM (Wavelength-Division Multiplexing)
    • OTN (Optical Transport Network)
      • OverHead of OTN
      • OTN Alarms & Errors
      • Do it again, what's happened?
  • Communication Tech English
    • Fundamentals of Electricity
    • Digital Communications
    • Optical Communications
  • High-speed railway Communication Technology
    • Overview
    • Base Knowledge
    • FH98
    • MDS3400
    • Everything is for the exam
  • GSM for Railway
    • Overview
    • Wired Parts
    • Digital dispatch Communication System
    • Basic Knowledge of GSM-R
    • Key technologies for GSM-R
    • Structure of GSM-R
    • GSM-R Network Mode
    • Wireless Channels for GSM-R
    • Mobility Management
    • Connection Management
    • Security Management
    • GPRS
    • GSM-R/GPRS Wireless Access Platform
    • GSM-R Features
    • GSM-R Numbering Plan
    • ASCI
  • Network Configuration Training
    • Words I have learned
  • Broadband Access Technology
    • Using Copper Line
    • Using Optical Fiber
    • Wireless
    • All for the test
  • CIR
    • Basci Knowledge
    • Testing Equipment
    • The Structure of CIR
    • All for the exam
  • LTE
  • Script for ChaoXing
  • Transmission and access network
Powered by GitBook
On this page
  • Reverse power rule
  • Indefinite integrals of , , and
  • Indefinite integral of
  • More trigonometric functions
  • Exponential functions

Was this helpful?

  1. High Level Math
  2. Integration
  3. Indefinite integral

Basic formulas

trigonometric 三角函数的

We call \sqrt{}​ radical.

Reverse power rule

∫xndx=xn+1n+1+C\int x^n dx = \frac{x^{n+1}}{n+1} + C∫xndx=n+1xn+1​+C

You increase the power by one and then divide by the power + 1

Remember that this rule doesn't apply for n=−1n = -1n=−1

∫xdx=∫x12dx=x12+112+1+C=x3232+C=x3⋅23+C=2x33+C\begin{align*} \int \sqrt{x} dx &= \int x^\frac{1}{2} dx \\ \\ & = \frac{x^{\frac{1}{2}+1}}{\frac{1}{2}+1} + C \\ \\ & = \frac{x^{\frac{3}{2}}}{\frac{3}{2}} + C \\ \\ & = \sqrt{x^3} \cdot \frac{2}{3} + C \\ \\ & = \frac{2\sqrt{x^3}}{3} + C \end{align*}∫x​dx​=∫x21​dx=21​+1x21​+1​+C=23​x23​​+C=x3​⋅32​+C=32x3​​+C​

Indefinite integrals of sin(x)sin(x)sin(x), cos(x)cos(x)cos(x), and exe^xex

∫sin⁡(x)dx=−cos⁡(x)+C∫cos⁡(x)dx=sin⁡(x)+C∫exdx=ex+C\begin{align*} \int \sin(x) dx &= -\cos(x) + C \\ \\ \int \cos(x) dx &= \sin(x) + C \\ \\ \int e^x dx &= e^x + C \end{align*}∫sin(x)dx∫cos(x)dx∫exdx​=−cos(x)+C=sin(x)+C=ex+C​
∫(sin⁡t+cos⁡t−et)dx=−cos⁡t+sin⁡t−et\begin{align*} \\ \\ &\int (\sin{t} + \cos{t} - e^t)dx \\ \\ &= -\cos{t} + \sin{t} - e^t \end{align*}​∫(sint+cost−et)dx=−cost+sint−et​

Indefinite integral of 1x\frac{1}{x}x1​

∫1xdx=ln⁡∣x∣+C\int \frac{1}{x} dx = \ln{|x|} + C∫x1​dx=ln∣x∣+C

More trigonometric functions

∫sec⁡2(x)dx=tan(x)+C∫sec⁡(x)tan⁡(x)dx=sec⁡(x)+C∫csc⁡2(x)dx=−cot⁡(x)+C∫csc⁡(x)cot⁡(x)dx=−csc⁡(x)+C\begin{align*} & \int \sec^2(x) dx = tan(x) + C \\ \\ & \int \sec(x) \tan(x) dx = \sec(x) + C \\ \\ & \int \csc^2(x) dx = -\cot(x) + C \\ \\ & \int \csc(x) \cot(x) dx = -\csc(x) + C \end{align*}​∫sec2(x)dx=tan(x)+C∫sec(x)tan(x)dx=sec(x)+C∫csc2(x)dx=−cot(x)+C∫csc(x)cot(x)dx=−csc(x)+C​

Exponential functions

∫axdx=axln⁡(a)+C\int a^x dx = \frac{a^x}{\ln(a)} + C∫axdx=ln(a)ax​+C
PreviousIndefinite integralNextUse formula

Last updated 6 years ago

Was this helpful?