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1.6 Integrals Involving Exponential and Logarithmic Functions

3 min read•june 24, 2024

Exponential and logarithmic functions are key players in calculus. They pop up everywhere, from compound interest to population growth. Mastering their integration techniques is crucial for tackling complex problems in math and science.

These functions have unique properties that make them both challenging and fascinating to work with. We'll explore how to handle their integrals, from basic exponential forms to tricky logarithmic expressions, and see how they apply to real-world scenarios.

Integration Techniques for Exponential and Logarithmic Functions

Integration of exponential functions

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  • Exponential functions have the form f(x)=axf(x) = a^x, where aa is a positive constant ([e](https://www.fiveableKeyTerm:e)[e](https://www.fiveableKeyTerm:e), 2, 10)
    • Derivative of axa^x is axln⁡aa^x \ln a using
    • of axa^x is axln⁡a+C\frac{a^x}{\ln a} + C found by reversing the derivative
  • Integrating exponential functions with variable exponents requires using specific rules:
    • ∫eaxdx=1aeax+C\int e^{ax} dx = \frac{1}{a}e^{ax} + C simple case with ee as the base (where ee is )
    • ∫xeaxdx=1a2(ax−1)eax+C\int x e^{ax} dx = \frac{1}{a^2}(ax - 1)e^{ax} + C linear term multiplied by exponential
    • ∫xneaxdx\int x^n e^{ax} dx can be solved using repeatedly for higher powers of xx
  • Examples of integrals demonstrate application:
    • ∫e3xdx=13e3x+C\int e^{3x} dx = \frac{1}{3}e^{3x} + C base ee with constant multiple of xx in exponent
    • ∫xe2xdx=14(2x−1)e2x+C\int x e^{2x} dx = \frac{1}{4}(2x - 1)e^{2x} + C linear term xx multiplied by exponential e2xe^{2x}

Integrals with logarithmic functions

  • function is denoted by ln⁡x\ln x (base ee)
    • Derivative of ln⁡x\ln x is 1x\frac{1}{x} using chain rule
    • Antiderivative of 1x\frac{1}{x} is ln⁡∣x∣+C\ln |x| + C found by reversing the derivative
  • Integrating logarithmic functions follows specific rules:
    • ∫ln⁡xdx=xln⁡x−x+C\int \ln x dx = x \ln x - x + C using integration by parts
    • ∫ln⁡(ax)dx=xln⁡(ax)−x+C\int \ln(ax) dx = x \ln(ax) - x + C with constant multiple aa inside logarithm
    • ∫1xdx=ln⁡∣x∣+C\int \frac{1}{x} dx = \ln |x| + C reciprocal of xx is the derivative of ln⁡x\ln x
  • Examples of logarithmic function integrals show how to apply the rules:
    • ∫ln⁡(2x)dx=xln⁡(2x)−x+C\int \ln(2x) dx = x \ln(2x) - x + C constant multiple inside logarithm
    • ∫13xdx=13ln⁡∣x∣+C\int \frac{1}{3x} dx = \frac{1}{3} \ln |x| + C reciprocal of linear term 3x3x
  • The can be used to integrate logarithms with different bases

Substitution for exponential and logarithmic integrals

  • Substitution simplifies integrals by changing the variable of integration
    • Substitution u=g(x)u = g(x) used when integrand contains a function and its derivative
    • After substituting, replace dxdx with dudu using the relationship du=g′(x)dxdu = g'(x) dx
  • Using substitution with exponential functions:
    • If integrand contains eg(x)g′(x)e^{g(x)} g'(x), let u=g(x)u = g(x) to simplify
    • Example: ∫ex22xdx\int e^{x^2} 2x dx, let u=x2u = x^2, then du=2xdxdu = 2x dx to convert integral
  • Using substitution with logarithmic functions:
    • If integrand contains g′(x)g(x)\frac{g'(x)}{g(x)}, let u=g(x)u = g(x) to simplify
    • Example: ∫1xln⁡xdx\int \frac{1}{x \ln x} dx, let u=ln⁡xu = \ln x, then du=1xdxdu = \frac{1}{x} dx to convert integral
  • After substituting:
    1. Integrate with respect to uu
    2. Substitute back to express the result in terms of the original variable xx

Applications in Differential Equations and Inverse Functions

  • Exponential and logarithmic integrals are crucial in solving
  • These integrals often appear when dealing with in calculus
  • Understanding these integration techniques is essential for modeling real-world phenomena in various fields

Key Terms to Review (27)

∫ 1/x dx: The integral of the function 1/x, which represents the natural logarithm function. This integral is a fundamental operation in calculus and is particularly relevant in the context of integrals involving exponential and logarithmic functions.
∫ ln x dx: The integral of the natural logarithm of x, or the indefinite integral of the natural logarithm function, is a fundamental operation in calculus that arises when evaluating integrals involving logarithmic functions. This term is particularly relevant in the context of integrals involving exponential and logarithmic functions, as covered in section 1.6 of the course material.
∫ ln(ax) dx: The integral $$ ext{∫ ln(ax) dx}$$ represents the process of finding the antiderivative of the natural logarithm function multiplied by a linear term. This integral is significant because it combines the properties of logarithmic functions with polynomial expressions, which often appear in various applications, such as physics and economics. Understanding how to evaluate this integral is essential for working with integrals involving exponential and logarithmic functions, especially in integration techniques like integration by parts.
∫ x e^{ax} dx: The expression ∫ x e^{ax} dx represents the integral of the product of a polynomial term, x, and an exponential function, e^{ax}. This integral is significant as it showcases how to integrate functions involving both polynomial and exponential components, which often appear in real-world applications such as physics and engineering.
∫ x^n e^{ax} dx: The expression ∫ x^n e^{ax} dx represents the integral of a polynomial function multiplied by an exponential function, where n is a non-negative integer and a is a constant. This type of integral often arises in applications involving growth and decay processes, as well as in solving differential equations. Understanding how to evaluate this integral is crucial since it combines techniques from both polynomial integration and exponential functions.
∫e^x dx: The integral of the exponential function e^x with respect to the variable x. This represents the antiderivative or indefinite integral of the exponential function, which is a fundamental operation in calculus involving exponential and logarithmic functions.
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.
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 Base Formula: The change of base formula is a mathematical rule that allows you to convert logarithms from one base to another. It is expressed as \( \log_b(a) = \frac{\log_k(a)}{\log_k(b)} \), where \( b \) is the original base, \( a \) is the argument, and \( k \) is the new base. This formula is crucial when dealing with logarithmic functions, especially in calculus, as it simplifies the process of integrating and differentiating logarithms with different bases.
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 Equations: Differential equations are mathematical equations that describe the relationship between a function and its derivatives. They are used to model and analyze various phenomena in science, engineering, and other fields where the rate of change of a quantity is of interest.
E: The number 'e' is an irrational constant approximately equal to 2.71828, and it serves as the base of the natural logarithm. It's crucial in mathematics, especially in calculus, because it naturally arises in processes involving growth and decay, making it essential for understanding exponential functions and their integrals.
Euler's number: Euler's number, denoted as 'e', is an irrational constant approximately equal to 2.71828. It serves as the base for natural logarithms and is critical in various mathematical contexts, particularly in calculus. This number arises naturally in the study of growth processes, compounding interest, and the behavior of exponential functions.
Exponential function: An exponential function is a mathematical expression in the form $$f(x) = a imes b^{x}$$, where 'a' is a constant, 'b' is a positive real number not equal to 1, and 'x' is the exponent. These functions exhibit rapid growth or decay, depending on the value of 'b', and are fundamental in modeling various natural phenomena such as population growth, radioactive decay, and financial interest. Their unique properties make them essential in calculus, particularly when dealing with integrals and series.
Exponential Integration: Exponential integration is the process of finding the antiderivative or indefinite integral of functions involving exponential terms. This type of integration is a fundamental technique in calculus, particularly when dealing with integrals involving exponential and logarithmic functions.
Fruit flies: Fruit flies are small insects commonly used in genetic experiments. They have rapid life cycles and simple genetic structures.
Growth of bacteria: The growth of bacteria typically follows an exponential pattern, which can be modeled using exponential functions. In calculus, this concept is crucial for solving integrals involving exponential growth and decay.
Improper Integrals: Improper integrals are a type of integral that involves infinite limits of integration or functions that are not defined at certain points within the interval of integration. They are an extension of the concept of definite integrals, allowing for the evaluation of integrals where the integrand becomes unbounded or the interval of integration is infinite.
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.
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.
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.
L'Hôpital's Rule: L'Hôpital's rule is a powerful technique used to evaluate limits of indeterminate forms, such as $0/0$ or $\infty/\infty$. It states that if the limit of a ratio of functions is an indeterminate form, then the limit can be found by taking the ratio of the derivatives of the numerator and denominator functions.
Ln: The natural logarithm, denoted as 'ln', is the logarithm to the base 'e', where 'e' is approximately equal to 2.71828. This function is essential in various mathematical applications, especially when dealing with exponential growth and decay, as it allows for the manipulation of exponential equations into linear form. The natural logarithm has unique properties that make it particularly useful in calculus, especially when integrating and differentiating functions involving exponential terms.
Logarithmic Differentiation: Logarithmic differentiation is a technique used to find the derivative of a function that involves logarithmic expressions. It involves transforming the original function by taking the natural logarithm of both sides, then differentiating the resulting expression, and finally applying the chain rule to obtain the derivative of the original function.
Natural Logarithm: The natural logarithm, denoted as $\ln(x)$, is a logarithmic function that represents the power to which the mathematical constant $e$ must be raised to get the value $x$. It is a fundamental concept in calculus, particularly in the study of integrals involving exponential and logarithmic functions, as well as in the analysis of exponential growth and decay processes.
Price–demand function: The price–demand function represents the relationship between the price of a good or service and the quantity demanded by consumers. It is typically expressed as $p = f(x)$, where $p$ is the price and $x$ is the quantity demanded.
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.
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Glossary
Glossary
1.7 Integrals Resulting in Inverse Trigonometric Functions