MindMap Gallery indefinite integral
This is a mind map about indefinite integrals. The main contents include: practice questions of indefinite integrals, application examples of indefinite integrals, problem-solving strategies of indefinite integrals, common types of indefinite integrals, basic properties of indefinite integrals, and Geometric meaning, the concept of primitive function.
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This is a mind map about the interpretation and summary of the relationship field e-book, Main content: Overview of the essence interpretation and overview of the relationship field e-book. "Relationship field" refers to the complex interpersonal network in which an individual influences others through specific behaviors and attitudes.
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indefinite integral
The concept of original function
Function F(x) is the original function of f(x)
F'(x) = f(x)
Pairs of functions that satisfy the derivative relationship
Symbolic representation of indefinite integrals
∫f(x)dx = F(x) C
C is the integral constant
The geometric meaning of indefinite integrals
The relationship between curve F(x) and curve f(x)
The graph of F(x) is the accumulation of the area under the graph of f(x)
Can be regarded as the sum of infinitesimal rectangular bars under the f(x) graph
Area function under the curve
F(x) represents the cumulative area from a certain point to point x
The area under a specific interval can be found by integrating
Basic properties of indefinite integrals
linear properties
The indefinite integral of the sum is equal to the sum of the indefinite integrals of the functions
∫f(x) g(x)dx = ∫f(x)dx ∫g(x)dx
The indefinite integral of a constant multiple is equal to the indefinite integral of a constant multiple of the original function.
∫k * f(x)dx = k * ∫f(x)dx, k is a constant
substitution method
First substitution method
Suppose ∫f(u)du = F(u) C, u = φ(x) has continuous derivatives, then ∫f[φ(x)]φ'(x)dx = ∫f[φ(x)]dφ( x) = F[φ(x)] C
Second substitution method
Integration by parts
Integration using the derivative rule of products
∫u dv = uv ∫v du
Choose appropriate u and dv to simplify the integration.
Priority: Against three fingers of power
Common types of indefinite integrals
Integrals of rational functions
A function whose numerator and denominator are both polynomials
Integrate term by term after decomposition into partial fractions
Processing when the degree of numerator is higher than the degree of denominator
Adding terms, subtracting terms or raising differential powers
Integrals of trigonometric functions
1 General method (universal substitution)
2 special methods (triangular deformation, substitution, division), several common substitution methods
If R(-sinx,cosx)=-R(sinx,cosx), let u=cosx, or make up dcosx
If R(sinx,-cosx)=-R(sinx,cosx), let u=sinx, or make up dsinx
If R(-sinx,-cosx)=R(sinx,cosx), let u=tanx, or make up dtanx
Integral of simple irrational functions
Problem-solving strategies for indefinite integrals
Determine the point type
Determine whether it is a basic point
Apply basic points table directly
Determine whether you need to exchange dollars or partial points
Choose an appropriate integration method based on the functional form
Simplify the integral expression
Simplify integrals by algebraic transformations
Such as factoring, rationalization, etc.
Simplify integrals by trigonometric transformations
Such as trigonometric identity conversion
Check the points result
Verify the derivative of the original function
Make sure the derivative is equal to the integrand
Check the applicability of the integration constant C
Ensure the universality of the integration results
Application examples of indefinite integrals
Applications in Economics
cost and benefit analysis
The integral of the cost function gives the total cost
The integral of the revenue function gives the total revenue
demand and supply analysis
The integral of the demand function gives the total demand
The integral of the supply function gives the total quantity supplied
Indefinite integral exercises
Basic question types
Integrals of simple polynomial functions
Practice basic integral skills
Integrals of basic trigonometric functions
Be familiar with the integration rules of trigonometric functions
Improve question types
Mixed integrals of complex polynomials and trigonometric functions
Practice integration by substitution and integration by parts
Integral of exponential and logarithmic functions
Deepen your understanding of integrals of exponential and logarithmic functions
Comprehensive application questions
Application of integrals in practical problems
Practice applying integrals to solve real-world problems
Integration strategy selection for complex functions
Practice choosing an appropriate integration method
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