5.6 Integrals Involving Exponential and Logarithmic Functions

Exponential and logarithmic functions are key players in calculus. They pop up in many real-world scenarios, from population growth to compound interest. Mastering their integration techniques is crucial for solving complex problems.

These functions have unique properties that make them stand out. Their integrals often involve themselves or their inverses, creating elegant solutions. Understanding these patterns helps simplify seemingly tricky integrals and builds a strong foundation for advanced calculus concepts.

Integration Techniques for Exponential and Logarithmic Functions

Integrals of exponential functions

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• Integrate the natural exponential function $e^x$ results in $e^x + C$, where $C$ is the constant of integration
• Integrate exponential functions with bases other than $e$ using the formula $\int a^x dx = \frac{a^x}{\ln a} + C$, where $a > 0$ and $a \neq 1$ ($a = 2, 10$)
• Multiply the result by the constant when integrating exponential functions multiplied by a constant $k$ ($k = 3, -5$)
  • $\int k \cdot e^x dx = k \cdot e^x + C$
  • $\int k \cdot a^x dx = \frac{k \cdot a^x}{\ln a} + C$
• Integrate exponential functions multiplied by a linear term $x$ using integration by parts
  • $\int x \cdot e^x dx = (x - 1) \cdot e^x + C$
  • $\int x \cdot a^x dx = \frac{x \cdot a^x}{\ln a} - \frac{a^x}{(\ln a)^2} + C$

Integration of logarithmic expressions

• Integrate the natural logarithm $\ln x$ using the formula $\int \ln x dx = x \ln x - x + C$
• Multiply the result by the constant when integrating natural logarithms multiplied by a constant $k$ ($k = 2, -3$)
  • $\int k \cdot \ln x dx = k \cdot (x \ln x - x) + C$
• Integrate logarithmic functions with bases other than $e$ using the change of base formula
  • $\int \log_a x dx = \frac{x \ln x - x}{\ln a} + C$, where $a > 0$ and $a \neq 1$ ($a = 2, 10$)
• Integrate logarithmic functions multiplied by a linear term $x$ using integration by parts
  • $\int x \cdot \ln x dx = \frac{x^2}{2} \ln x - \frac{x^2}{4} + C$
  • $\int x \cdot \log_a x dx = \frac{x^2 \ln x - \frac{x^2}{2}}{\ln a} + C$

Substitution for exponential and logarithmic integrals

• Apply substitution to exponential functions by letting $u$ equal the exponent and adjusting $du$ and $dx$ accordingly
  1. Let $u = 2x$, then $du = 2dx$ or $dx = \frac{du}{2}$
  2. Rewrite the integral in terms of $u$ and simplify
  3. Integrate with respect to $u$ and substitute back the original variable
     • $\int e^{2x} \cdot 2 dx = \int e^u \cdot \frac{du}{2} = \frac{1}{2} \int e^u du = \frac{1}{2} e^u + C = \frac{1}{2} e^{2x} + C$
• Apply substitution to logarithmic functions by letting $u$ equal the argument of the logarithm and adjusting $du$ and $dx$
  1. Let $u = 3x$, then $du = 3dx$ or $dx = \frac{du}{3}$
  2. Rewrite the integral in terms of $u$ and simplify
  3. Integrate with respect to $u$ and substitute back the original variable
     • $\int \ln(3x) dx = \int \ln u \cdot \frac{du}{3} = \frac{1}{3} \int \ln u du = \frac{1}{3} (u \ln u - u) + C = \frac{1}{3} (3x \ln(3x) - 3x) + C$
• Recognize compositions of exponential and logarithmic functions that simplify the integral
  • $\int e^{\ln x} dx = \int x dx = \frac{x^2}{2} + C$, since $e^{\ln x} = x$

Fundamental Concepts of Integration

• Indefinite integral represents a family of antiderivatives differing by a constant
• Definite integral calculates the area under a curve between two points
• Fundamental Theorem of Calculus connects differentiation and integration, allowing us to evaluate definite integrals using antiderivatives
• Exponential function (e.g., $e^x$) and logarithmic function (e.g., $\ln x$) are inverse functions, which is useful when integrating composite functions