24 Questions
What is the first step in solving a second-order differential equation?
Let dy/dx = V and (d^2 y)/(dx^2) = dv/dx
What is the purpose of applying boundary conditions in solving a differential equation?
To find the particular solution of the differential equation
What is the resulting expression for V in the example of the second-order differential equation dv/dx = 4v?
V = e^4x
What is the general form of a second-order differential equation?
d^2 y/dx^2 = f(x, y)
What is the final answer for the example of the first-order differential equation dy/dx = xy^2 + 9x?
y = 3 tan^(-1)((3x^2)/2 - 23π/4)
What is the purpose of rearranging the equation to find y exclusively in solving a differential equation?
To express the dependent variable in terms of the independent variable
What does the 'I' in the InLATE rule stand for?
Inverse Trigonometric Functions
What is the order of preference when choosing u and dv in integration by parts using the InLATE rule?
Inverse Trigonometric Functions, Natural Logarithms, Logarithms, Algebraic Functions, Trigonometric Functions, Exponential Functions
What is the first step in solving a differential equation?
Separate the variables
What is the purpose of the InLATE rule in integration by parts?
To choose u and dv in integration by parts
What is the final step in solving a differential equation?
Rearrange to the correct form
What is the purpose of solving a differential equation?
To get the variable on top in terms of the one on the bottom
What does the 'T' in the InLATE rule stand for?
Trigonometric Functions
What is the correct order of operations when using integration by parts?
Differentiate u, integrate dv, substitute into the formula
What is the purpose of integration by parts?
To integrate products of functions
What is the correct form of the solution to a differential equation?
y = something with x's
What is the integral of x^n with respect to x when n is not equal to 1?
x^(n+1)/(n+1)
What is the integral of 1/x with respect to x?
ln|x|
What is the integral of e^x with respect to x?
e^x + c
What is the key point to consider when choosing what to substitute in integration by substitution?
Look for something that is the derivative of another part of the function.
What is the formula for integration by parts?
∫u dv = uv - ∫v du
What is the integral of 2x e^(x^2) with respect to x?
e^(x^2) + c
When using integration by parts, what is the first step?
Identify u and dv
What is the purpose of integration by substitution?
To make integration easier by replacing a complex part with a simpler variable
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
Practice exercises on rules of integration in calculus, including power rule, exponential and trigonometric functions.
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