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1.5 Substitution

2 min read•june 24, 2024

The substitution method is a powerful tool for simplifying complex integrals. By changing variables, we can transform tricky integrands into more manageable forms. This technique is especially handy when dealing with or expressions that resemble the .

Substitution works for both indefinite and , with slight variations in the process. Recognizing when to use substitution is key - look for composite functions or inner functions paired with their derivatives. It's a versatile method that often paves the way for solving challenging integration problems.

Substitution Method for Integration

Substitution for indefinite integrals

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  • Simplifies and evaluates integrals by changing the variable of integration
  • Particularly useful for integrands that are composite functions (function inside another function)
  • Steps to apply substitution:
    1. Identify part of integrand to replace with new variable uu
    2. Determine differential of uu (dudu) by differentiating uu with respect to original variable
    3. Replace identified part of integrand with uu and corresponding differential with dudu
    4. Simplify new integrand and evaluate integral with respect to uu
    5. Substitute back original expression for uu to obtain in terms of original variable
  • Substitution is closely related to the chain rule for derivatives, but applied in reverse

Substitution in definite integrals

  • Substitution also applies to definite integrals
  • Additional steps when using substitution in definite integrals:
    • Change to correspond to new variable uu
      • Evaluate substitution uu at original lower and upper limits to find new limits
    • Evaluate definite integral with respect to uu using new limits
    • Substitute back original expression for uu to obtain final result
  • The is applied after substitution to evaluate the definite integral

Recognition of substitution-suitable integrands

  • Characteristics of integrands suitable for substitution:
    • Integrand is a composite function
    • Inner function appears along with its derivative multiplied by some other term
  • Common substitutions:
    • (u=sin⁡xu = \sin x, u=cos⁡xu = \cos x, u=tan⁡xu = \tan x)
    • (u=exu = e^x, u=axu = a^x)
    • (u=ln⁡xu = \ln x, u=log⁡axu = \log_a x)
    • (u=xu = \sqrt{x}, u=ax+bu = \sqrt{ax + b})
    • (u=ax+bu = ax + b, u=ax2+bx+cu = ax^2 + bx + c)
  • Choice of substitution depends on form of integrand and aim to simplify integral
  • Substitution should result in simpler integrand that can be easily integrated using known techniques

Advanced Integration Techniques

  • : Used when the integrand is a product of functions
  • : Substitution can be particularly useful when dealing with integrals involving inverse functions
  • : The use of dudu in substitution is an example of differential notation, which helps visualize the substitution process
  • : A more general form of substitution used in multivariable calculus

Key Terms to Review (25)

Antiderivative: An antiderivative, also known as a primitive function or indefinite integral, is a function whose derivative is the original function. It represents the accumulation or the reverse process of differentiation, allowing us to find the function that was differentiated to obtain a given derivative.
Back-substitution: Back-substitution is a method used to solve systems of equations, particularly in the context of solving integrals through substitution. After performing a substitution to simplify the integral, back-substitution involves replacing the variable in the resulting expression with its original expression to find the solution in terms of the original variable. This process is essential in ensuring that the final answer aligns with the variables and limits of the original problem.
Chain Rule: The chain rule is a fundamental concept in calculus that allows for the differentiation of composite functions. It provides a systematic way to find the derivative of a function that is composed of other functions.
Change of variables: Change of variables is a method used to simplify integrals by substituting a new variable for an existing one. This technique often makes the integral easier to evaluate.
Change of Variables: Change of variables, also known as substitution, is a technique used in calculus to simplify the integration of complex functions by transforming the original variable into a new variable. This method allows for the conversion of an integral with a complicated integrand into an integral with a simpler integrand, making it easier to evaluate.
Composite Functions: Composite functions are a type of function where the output of one function becomes the input for another function. This combination of functions allows for more complex mathematical operations and transformations to be performed on data or variables.
Definite Integrals: A definite integral is a mathematical operation that calculates the area under a curve on a graph between two specific points. It represents the accumulation of a quantity over an interval and is a fundamental concept in calculus that connects the ideas of differentiation and integration.
Differential Notation: Differential notation is a mathematical representation used to express infinitesimal changes or derivatives in calculus. It is a concise way of denoting the rate of change of a variable with respect to another variable, which is a fundamental concept in the study of derivatives and integrals.
Du/dx: The term du/dx represents the derivative of the variable u with respect to the variable x. It plays a crucial role in calculus, particularly in the context of substitution, where it helps transform complex integrals into simpler ones by changing variables. Understanding du/dx allows for easier manipulation of integrals, making it essential for applying the substitution method effectively.
Exponential Functions: Exponential functions are mathematical expressions of the form $$f(x) = a imes b^{x}$$ where 'a' is a constant, 'b' is the base (a positive real number), and 'x' is the exponent. These functions model rapid growth or decay and are essential in various applications, such as compound interest and population growth, due to their unique property where the rate of change is proportional to the function's current value.
Fundamental Theorem of Calculus: The Fundamental Theorem of Calculus is a central result in calculus that establishes a deep connection between the concepts of differentiation and integration. It provides a powerful tool for evaluating definite integrals and understanding the relationship between the rate of change of a function and the function itself.
Indefinite integrals: An indefinite integral, also known as an antiderivative, is a function that reverses the process of differentiation. It represents a family of functions whose derivative is the given function.
Indefinite Integrals: Indefinite integrals represent a family of functions whose derivatives yield the original function, encapsulating the concept of anti-differentiation. They are expressed with the integral sign followed by a function and the differential, and they include a constant of integration, indicating that there are infinitely many functions that differ by a constant. This idea is central to techniques like substitution and trigonometric identities, allowing for more complex expressions to be simplified and solved.
Integration by Parts: Integration by parts is a technique used to integrate products of functions by transforming the integral into a simpler form using the formula $$\int u \, dv = uv - \int v \, du$$. This method connects various integration strategies, making it especially useful in situations where other techniques like substitution may not be effective.
Integration by substitution: Integration by substitution is a method for finding integrals by making a substitution to simplify the integral. It involves changing variables to rewrite an integral in a simpler form.
Inverse functions: Inverse functions are pairs of functions that essentially 'undo' each other, meaning that if you apply one function and then its inverse, you will return to your original input. For any function \( f(x) \), its inverse \( f^{-1}(x) \) satisfies the condition \( f(f^{-1}(x)) = x \) for all x in the domain of \( f^{-1} \). Understanding inverse functions is crucial for solving equations, particularly when dealing with logarithmic and exponential forms.
Limits of Integration: Limits of integration are the specified bounds that define the interval over which an integral is evaluated. They indicate the starting and ending points of the integration process, allowing for the calculation of areas, volumes, or accumulated quantities within that range. These limits can be constants or variable expressions and play a crucial role in determining the results of definite integrals.
Logarithmic functions: Logarithmic functions are the inverse of exponential functions and are defined as $$y = ext{log}_b(x)$$, where $$b$$ is the base, and $$x$$ is a positive number. They play a crucial role in many mathematical applications, particularly in solving equations involving exponents and in integration techniques. Understanding logarithmic functions is essential for performing substitutions in integrals, applying integration by parts effectively, and employing various other strategies for integration to simplify complex problems.
Maclaurin polynomials: Maclaurin polynomials are special cases of Taylor polynomials centered at $x = 0$. They provide polynomial approximations of functions using derivatives evaluated at zero.
Polynomials: Polynomials are algebraic expressions composed of variables and coefficients, where the variables are raised to non-negative integer powers. They are fundamental mathematical objects that play a crucial role in various areas of calculus, including substitution and integration techniques.
Square Roots: A square root is a value that, when multiplied by itself, gives the original number. It is represented by the radical symbol '√' and plays a crucial role in simplifying expressions and solving equations, especially in the context of substitutions in calculus. Understanding square roots is essential for manipulating algebraic expressions and applying techniques such as trigonometric substitution.
The Integral Symbol (∫): The integral symbol (∫) represents the mathematical operation of integration, which is the inverse of differentiation. It is used to calculate the accumulated change of a function over an interval, finding the area under a curve, or determining the total effect of a varying quantity.
Trigonometric Functions: Trigonometric functions are mathematical functions that describe the relationship between the angles and sides of a right triangle. They are fundamental in the study of calculus and are essential in understanding integration formulas, substitution, and other integration strategies.
U-substitution: U-substitution is a technique used in integration that simplifies the process by substituting a part of the integral with a new variable, usually denoted as 'u'. This method allows for easier integration by transforming complex expressions into simpler ones, facilitating the calculation of definite and indefinite integrals.
Variable Transformation: Variable transformation is a technique used in calculus to simplify and solve integrals or differential equations by changing the variable of integration or differentiation. This process involves substituting a new variable in place of the original variable, which can make the integral or equation easier to evaluate.
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Glossary
Glossary
1.6 Integrals Involving Exponential and Logarithmic Functions