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Magazine Chapter 7: Problem 29
Perform the indicated integrations. \(\int e^{x} \sec e^{x} d x\)
Short Answer
Expert verified
The integral is \( \ln |\sec(e^x) + \tan(e^x)| + C \).
Step by step solution
01
Identify the substitution
To solve the integral \( \int e^{x} \sec e^{x} \, dx \), we look for a substitution that simplifies the expression. Notice that the integral involves \( e^x \) both as a factor and as an argument of the secant function, suggesting a substitution of \( t = e^x \).
02
Differentiate the substitution
Differentiate the substitution with respect to \( x \):\[ dt = e^x \, dx \]This expression allows us to replace \( e^x \, dx \) with \( dt \) in the integral.
03
Rewrite the integral using substitution
Substitute \( t = e^x \) and \( dt = e^x \, dx \) back into the integral:\[ \int \sec t \, dt \]
04
Integrate the transformed integral
The integral \( \int \sec t \, dt \) is a standard integral with the result:\[ \int \sec t \, dt = \ln |\sec t + \tan t| + C \]where \( C \) is the constant of integration.
05
Substitute back for the original variable
Substitute back \( t = e^x \) to express the solution in terms of \( x \):\[ \ln |\sec(e^x) + \tan(e^x)| + C \]
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Integral Substitution
Integral substitution is a powerful technique in calculus for simplifying integrals, especially when faced with complex expressions. It involves changing the variable of integration to make the integral more manageable. Here's a step-by-step explanation:
- First, identify a part of the integral that can be transformed into a simpler form. This choice is often guided by a repeated expression or a function's derivative. For example, in \( \int e^{x} \sec e^{x} \, dx \), notice that both a function \( e^x \) and its derivative appear together, making it an excellent candidate for substitution.
- Next, introduce a new variable to replace the complex expression. In our case, by setting \( t = e^x \), we simplify the integral's expression.
- Differentiating \( t = e^x \) with respect to \( x \) gives \( dt = e^x \, dx \). This allows you to replace terms in the integral with respect to \( t \) as follows: the integral \( \int e^{x} \sec e^{x} \, dx \) becomes \( \int \sec t \, dt \).
Exponential Functions
Exponential functions are fundamental in mathematics with many applications in science and engineering. The general form of an exponential function is \( f(x) = a \cdot e^{b x} \), where \( e \) is Euler's number, approximately 2.71828. Here's how exponential functions are used:
- They model growth and decay processes such as population growth, radioactive decay, and cooling laws. These situations all exhibit constant proportionality in their change rates.
- In calculus, derivatives and integrals involving exponential functions can often simplify the solution process. Specifically, the derivative of \( e^x \) is itself, making it incredibly straightforward to work with.
- When integrating functions like \( \int e^{x} \sec e^{x} \, dx \), recognizing the role of \( e^x \) helps guide the use of substitution, linking the integral terms through substitution like \( t = e^x \).
Trigonometric Integrals
Integrals involving trigonometric functions often require special techniques or substitutions because they involve non-algebraic expressions. Here are some key points to consider when dealing with them:
- Recognize standard integrals. For instance, knowing that \( \int \sec t \, dt = \ln |\sec t + \tan t| + C \) helps quickly solve integrals once reduced to standard form.
- Substitution is crucial. Often, trigonometric integrals require rewriting expressions using identities or substitutions to simplify our calculations. This method is highlighted by our initial integral transformation.
- Understand the relationship between trigonometric functions and their derivatives. Solving these integrals sometimes involves recognizing derivative patterns. For example, derivative behavior might suggest using conjugate trigonometric functions.
