Calculus: Rules of Integration

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Questions and Answers

What is the first step in solving a second-order differential equation?

Separate the variables and integrate

Apply the boundary conditions

Let dy/dx = V and (d^2 y)/(dx^2) = dv/dx (correct)

Rearrange the equation to find y exclusively

What is the purpose of applying boundary conditions in solving a differential equation?

To find the particular solution of the differential equation (correct)

To determine the order of the differential equation

To find the general solution of the differential equation

To separate the variables and integrate

What is the resulting expression for V in the example of the second-order differential equation dv/dx = 4v?

V = 4x + c

V = x^2 + c

V = e^4x (correct)

V = 4e^4x

What is the general form of a second-order differential equation?

d^2 y/dx^2 = f(x, y) Signup and view all the answers

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) Signup and view all the answers

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 Signup and view all the answers

What does the 'I' in the InLATE rule stand for?

Inverse Trigonometric Functions Signup and view all the answers

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 Signup and view all the answers

What is the first step in solving a differential equation?

Separate the variables Signup and view all the answers

What is the purpose of the InLATE rule in integration by parts?

To choose u and dv in integration by parts Signup and view all the answers

What is the final step in solving a differential equation?

Rearrange to the correct form Signup and view all the answers

What is the purpose of solving a differential equation?

To get the variable on top in terms of the one on the bottom Signup and view all the answers

What does the 'T' in the InLATE rule stand for?

Trigonometric Functions Signup and view all the answers

What is the correct order of operations when using integration by parts?

Differentiate u, integrate dv, substitute into the formula Signup and view all the answers

What is the purpose of integration by parts?

To integrate products of functions Signup and view all the answers

What is the correct form of the solution to a differential equation?

y = something with x's Signup and view all the answers

What is the integral of x^n with respect to x when n is not equal to 1?

x^(n+1)/(n+1) Signup and view all the answers

What is the integral of 1/x with respect to x?

ln|x| Signup and view all the answers

What is the integral of e^x with respect to x?

e^x + c Signup and view all the answers

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. Signup and view all the answers

What is the formula for integration by parts?

∫u dv = uv - ∫v du Signup and view all the answers

What is the integral of 2x e^(x^2) with respect to x?

e^(x^2) + c Signup and view all the answers

When using integration by parts, what is the first step?

Identify u and dv Signup and view all the answers

What is the purpose of integration by substitution?

To make integration easier by replacing a complex part with a simpler variable Signup and view all the answers

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

Differential equations involve derivatives; to solve them, separate variables, set up integration, and integrate both sides, then rearrange to the correct form.