Rules of Integration and Substitution
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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?

<p>To find the value of the constant <code>c</code> and ensure the solution satisfies the initial conditions</p> Signup and view all the answers

If dv/dx=4v, how can we solve for v?

<p>Separate the variables as <code>dv/4v=dx</code> and integrate to get <code>1/4 ln⁡4v=x+c</code></p> Signup and view all the answers

What is the next step after solving for v in a second-order differential equation?

<p>Substitute <code>v</code> back into the original equation <code>dy/dx=v</code> and solve for <code>y</code></p> Signup and view all the answers

What is the general power rule of integration with constants of integration omitted, and what is the exception to this rule?

<p>The general power rule of integration is ∫x^n dx = x^(n+1)/(n+1), and the exception is when n = 1.</p> Signup and view all the answers

What is the integral of 1/x dx, and how does it differ from the integral of 1/(ax+b) dx?

<p>The integral of 1/x dx is ln|x|, while the integral of 1/(ax+b) dx is 1/a ln|ax+b|.</p> Signup and view all the answers

How do you choose the substitution in integration by substitution, and what is the key point to keep in mind?

<p>Choose a part of the integrand whose derivative is also present, and look for something that is the derivative of another part of the function (even with different coefficients).</p> Signup and view all the answers

What is the formula for integration by parts, and how is it based on the product rule of differentiation?

<p>The formula for integration by parts is ∫u dv = uv - ∫v du, which is based on the product rule of differentiation.</p> Signup and view all the answers

Evaluate the integral ∫(2x+1) dx. Show your work.

<p>∫(2x+1) dx = x^2 + x + C</p> Signup and view all the answers

Find the integral ∫_34^5 (1/(5x+1)) dx. Show your work.

<p>∫_34^5 (1/(5x+1)) dx = (1/5) ln|5x+1| |_34^5 = (1/5) ln|25| - (1/5) ln|19|</p> Signup and view all the answers

Evaluate the integral ∫_(-π)^π cos(x) dx. Show your work.

<p>∫<em>(-π)^π cos(x) dx = sin(x) |</em>(-π)^π = sin(π) - sin(-π) = 0</p> Signup and view all the answers

Use integration by substitution to evaluate the integral ∫(3x^2)/√(x^3+3) dx from 1 to 4. Show your work.

<p>Let u = x^3 + 3, then du/dx = 3x^2. Substituting, we get ∫(3x^2)/√(x^3+3) dx = ∫(1/√u) du = 2√u + C. Evaluating the definite integral, we get 2√(4^3+3) - 2√(1^3+3) = 2√67 - 2√4.</p> Signup and view all the answers

What does the InLATE rule stand for in integration by parts?

<p>Inverse Trigonometric Functions, Natural Logarithms, Logarithms, Algebraic Functions, Trigonometric Functions, and Exponential Functions</p> Signup and view all the answers

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?

<p>Inverse Trigonometric Functions</p> Signup and view all the answers

What is the formula for integrating by parts?

<p>∫udv = uv - ∫vdu</p> Signup and view all the answers

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

<p>To choose which part of the integrand to differentiate (du) and which part to integrate (dv)</p> Signup and view all the answers

What is the process of solving a differential equation?

<p>Separate the variables, set up integration and choose limits, integrate both sides, and rearrange to the correct form</p> Signup and view all the answers

What does 'solve' mean in the context of differential equations?

<p>Get the variable on top in terms of the one on the bottom</p> Signup and view all the answers

What is the first step in solving a differential equation?

<p>Separate the variables</p> Signup and view all the answers

What is the purpose of integrating both sides in solving a differential equation?

<p>To get the general solution of the differential equation</p> Signup and view all the answers

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

<p>To find the particular solution of the differential equation</p> Signup and view all the answers

What is the relationship between integration by parts and differential equations?

<p>Both involve calculus and integration</p> 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

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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.

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