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Chapter 5: Problem 21

Evaluate the integrals using appropriate substitutions. $$\int e^{2 x} d x$$

Short Answer

Expert verified
The integral is \( \frac{1}{2} e^{2x} + C \).

Step by step solution

01

Identify the Substitution

To simplify the integral, we use the substitution method. Let's choose \( u = 2x \).
02

Differentiate the Substitution

Find the derivative of \( u \) with respect to \( x \), giving \( \frac{du}{dx} = 2 \), so \( dx = \frac{du}{2} \).
03

Rewrite the Integral with the Substitution

Express the original integral in terms of \( u \). The integral becomes \( \int e^u \cdot \frac{du}{2} \).
04

Integrate

Factor out the constant \( \frac{1}{2} \) from the integral and solve it: \( \frac{1}{2} \int e^u \, du \). The integral of \( e^u \) is \( e^u \), so the result becomes \( \frac{1}{2} e^u + C \).
05

Substitute Back

Replace \( u \) with \( 2x \) to get back to the variable \( x \). The final result is \( \frac{1}{2} e^{2x} + C \).

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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 tool in calculus used to simplify integrals and make them easier to evaluate. It involves selecting a new variable to replace an existing expression in the integral, which often results in a simpler integral to solve. In this method, you follow these basic steps:
  • Identify a part of the integral to substitute with a new variable.
  • Differentiate the substitution to express the differential in terms of the new variable.
  • Rewrite the entire integral with this new substitution.
  • Integrate with respect to the new variable.
  • Finally, replace the new variable back with the original expressions.
In the original problem, we chose the substitution \( u = 2x \). By differentiating, we found \( dx \) in terms of \( du \), simplifying our original expression \( \int e^{2x} \, dx \) to \( \int e^u \frac{du}{2} \). This substitution transforms a complex exponential integral into a basic one that is much easier to solve.
Exponential Functions
Exponential functions are mathematical functions of the form \( f(x) = a^x \), where \( a \) is a positive constant. In calculus, we often deal with natural exponential functions, written as \( e^x \), where \( e \approx 2.71828 \) is the base of the natural logarithm. These functions have unique properties:
  • The rate of change of \( e^x \) is proportional to its value, which means its derivative is also \( e^x \).
  • This property simplifies differentiation and integration, as the integral of \( e^x \) is also \( e^x \).
  • Exponential growth and decay processes in real-life applications often use \( e^x \).
In the context of indefinite integrals, knowing that the integral of an exponential function \( e^u \) is itself \( e^u \) allows us to easily compute integrals involving exponential expressions. Substituting back the variable after integration, as we did with \( u = 2x \), results in functions like \( \frac{1}{2} e^{2x} + C \) in the solution.
Indefinite Integrals
Indefinite integrals are the reverse process of differentiation, aiming to find a function whose derivative matches the given function. Unlike definite integrals which calculate the area under a curve for a specific interval, indefinite integrals represent a family of functions and include a constant of integration, denoted by \( C \).
  • Indefinite integrals are represented by the integral symbol \( \int \) without limits, followed by a function and differential, like \( \int f(x) \, dx \).
  • The result of an indefinite integral is a family of functions plus an arbitrary constant (\( C \)).
  • The constant \( C \) reflects the fact that there are infinitely many antiderivatives corresponding to the vertical shift of the curve.
In the exercise, we evaluated the indefinite integral \( \int e^{2x} \, dx \) and derived \( \frac{1}{2} e^{2x} + C \) as the antiderivative, illustrating how constants play a crucial role in the broad family of potential solutions.

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