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Questions and Answers
What is the general form of the antiderivative of x^n?
x^(n+1)/(n+1)
What is the strategy for solving second-order differential equations?
Let dy/dx=V and (d^2 y)/(dx^2)=dv/dx, then solve the resulting differential equation for V, and finally solve for y.
If dy/dx=xy^2+9x, what substitution can be used to separate the variables?
Factor out x and separate the variables as dy/(y^2+9)=x dx
What is the purpose of applying boundary conditions when solving differential equations?
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If dv/dx=4v, how can we solve for v?
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What is the next step after solving for v in a second-order differential equation?
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What is the general power rule of integration with constants of integration omitted, and what is the exception to this rule?
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What is the integral of 1/x dx, and how does it differ from the integral of 1/(ax+b) dx?
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How do you choose the substitution in integration by substitution, and what is the key point to keep in mind?
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What is the formula for integration by parts, and how is it based on the product rule of differentiation?
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Evaluate the integral ∫(2x+1) dx. Show your work.
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Find the integral ∫_34^5 (1/(5x+1)) dx. Show your work.
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Evaluate the integral ∫_(-π)^π cos(x) dx. Show your work.
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Use integration by substitution to evaluate the integral ∫(3x^2)/√(x^3+3) dx from 1 to 4. Show your work.
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What does the InLATE rule stand for in integration by parts?
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What is the first preference in the InLATE rule when deciding which part of the integrand to choose as u and which part as dv?
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What is the formula for integrating by parts?
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What is the purpose of the InLATE rule in integration by parts?
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What is the process of solving a differential equation?
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What does 'solve' mean in the context of differential equations?
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What is the first step in solving a differential equation?
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What is the purpose of integrating both sides in solving a differential equation?
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What is the role of boundary conditions in solving a differential equation?
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What is the relationship between integration by parts and differential equations?
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Study Notes
Rules of Integration
- ∫x^n dx = x^(n+1)/(n+1) for n ≠ 1
- ∫(1/x) dx = ln|x|
- ∫(1/(ax+b)) dx = (1/a) ln|(ax+b)|
- ∫e^x dx = e^x
- ∫e^(ax+b) dx = (1/a) e^(ax+b)
- ∫sin(x) dx = -cos(x)
- ∫sin(ax+b) dx = (-1/a) cos(ax+b)
- ∫cos(x) dx = sin(x)
- ∫cos(ax+b) dx = (1/a) sin(ax+b)
Integration by Substitution
- A helpful technique for difficult functions
- Replace a complex part with a simpler variable to make it easier to integrate
- Key point: Choose a part to substitute that is the derivative of another part of the function
Integration by Parts
- A technique for integrating products of two functions
- Based on the product rule for differentiation
- Formula: ∫u dv = uv - ∫v du
- Steps:
- Identify u and dv using the InLATE rule
- Differentiate u to find du
- Integrate dv to find v
- Substitute into the formula
- InLATE rule:
- I: Inverse Trigonometric Functions
- n: Natural Logarithms
- L: Logarithms
- A: Algebraic Functions
- T: Trigonometric Functions
- E: Exponential Functions
Differential Equations
- Equations involving derivatives (dy/dx, etc.)
- To solve differential equations, use calculus
- Steps:
- Separate the variables
- Set up integration and choose limits
- Integrate both sides and calculate limits
- Rearrange to the correct form
- Example: dy/dx = x^2, with y = 1 when x = 2
Second Order Differential Equations
- Equations that include (d^2 y)/(dx^2)
- Steps to solve:
- Let dy/dx = V and (d^2 y)/(dx^2) = dv/dx
- Solve the resulting differential equation to get an expression for V
- Let this expression = dy/dx and solve again
- Example: dv/dx = 4v
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Description
This quiz covers the rules of integration, including power rule, logarithmic, exponential, and trigonometric functions, as well as integration by substitution.
