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Magazine Chapter 5: Problem 65
Find a formula for \(\int e^{a x+b} d x \quad\) where \(a\) and \(b\) are constants.
Short Answer
Expert verified
The formula is \( \int e^{ax+b} \, dx = \frac{1}{a} e^{ax+b} + C \).
Step by step solution
01
Identify the Structure of the Integral
The integral given is of the form \( \int e^{u} \, du \), where \( u = ax + b \). This suggests the use of substitution to simplify the integration process.
02
Perform Substitution
Set \( u = ax + b \). Then, differentiate \( u \) with respect to \( x \) to obtain \( \frac{du}{dx} = a \), or equivalently \( du = a \, dx \). Thus, \( dx = \frac{du}{a} \). Substitute these into the integral.
03
Substitute and Integrate
Substitute \( u \) and \( du \) into the integral: \( \int e^{ax+b} \, dx = \int e^{u} \frac{du}{a} \). This simplifies to \( \frac{1}{a} \int e^{u} \, du \). The integral of \( e^{u} \) with respect to \( u \) is \( e^{u} \). Therefore, \( \int e^{u} \frac{du}{a} = \frac{1}{a} e^{u} + C \), where \( C \) is an integration constant.
04
Substitute Back to Original Variable
Now, replace \( u \) with its original expression \( ax + b \). Therefore, the integral becomes \( \frac{1}{a} e^{ax+b} + C \).
05
Write the Solution
The formula for \( \int e^{ax+b} \, dx \) is \( \frac{1}{a} e^{ax+b} + C \). This is the result of the integration performed with substitution.
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Substitution Method
The substitution method is a powerful technique used to simplify the integration of certain functions. It involves replacing a complex part of an integrand with a single variable, which can make the problem easier to solve. The process includes the following steps:
- Identify the Part to Substitute: Look for a part of the integrand that, when replaced by a new variable, simplifies the integral. This is often a compound expression like a function within another function.
- Perform the Substitution: Define a new variable, commonly denoted as \( u \), which equals the identified expression. For example, in the given problem, we set \( u = ax + b \).
- Differentiate the Substitution: Determine \( du \) by differentiating \( u \) with respect to \( x \). This helps in redefining \( dx \) in terms of \( du \).
- Rewrite the Integral: Replace all instances of the original expression and \( dx \) in the integral with \( u \) and \( du \).
- Integrate and Substitute Back: After integrating in terms of \( u \), substitute back the original variable to present the final result.
Exponential Functions
Exponential functions are a fundamental type of mathematical function that involves steady growth or decay. They take the form of \( e^{ax+b} \), where \( e \) (approximately 2.718) is the base of the natural logarithm, and \( a \) and \( b \) are constants. These functions have several properties that make them crucial, particularly in calculus:
- Nature of Exponential Growth: In exponential functions, the value increases or decreases rapidly. The function's rate of change is proportional to its current value, making it significant in fields such as biology, finance, and physics.
- Derivative and Integral: The derivative of an exponential function is proportional to the exponential function itself. Similarly, the integral of an exponential function, such as \( e^{ax+b} \), involves substitution techniques to simplify to a basic form of \( e^u \).
- Applications: Exponential functions model a wide range of real-world systems including population growth, radioactive decay, and interest compounding. Understanding their behavior is essential to interpreting various natural and financial phenomena.
Definite and Indefinite Integrals
Integrals are a core concept in calculus, and they come in two main types: definite and indefinite. Understanding these two forms is key to solving integration problems.
- Indefinite Integrals: An indefinite integral represents a family of functions and includes a constant of integration, \( C \). It is used to find the antiderivative or the general form of the original function. For example, the solution \( \int e^{ax+b} \, dx = \frac{1}{a} e^{ax+b} + C \) is an indefinite integral.
- Definite Integrals: Conversely, a definite integral has upper and lower limits and results in a number. It represents the accumulated quantity, such as area under a curve, between two points. It's expressed as \( \int_{a}^{b} f(x) \, dx \).
- Difference in Applications: While indefinite integrals provide a range of potential solutions, definite integrals give a precise value for a given interval. Indefinite integrals are used in solving differential equations, while definite integrals are essential for calculating areas, volumes, and other quantities that need precise computation.
