Problem 57 - Calculus Early Transcendentals

Question

Additional integrals Evaluate the following integrals. $$\int e^{x} \sec \left(e^{x}+1\right) d x$$

Step-by-Step Solution

Answer

Question: Evaluate the integral of the function $$\int e^{x} \sec \left(e^{x}+1\right) d x$$. Answer: The integral of the given function is $$\int e^{x} \sec \left(e^{x}+1\right) d x = \ln|\sec(e^x+1) + \tan(e^x+1)| + C$$.

Step 1: Choose a substitution

Let us use the substitution method to solve this integral. We observe that it could be easier to compute if we let: $$u = e^x + 1$$ Hence, we have to find the derivative of \(u\) with respect to \(x\), which is: $$\frac{du}{dx} = \frac{d}{dx} \left( e^x + 1\right)$$

Step 2: Find the derivative of u

Differentiating \(u\) with respect to \(x\), we get: $$\frac{du}{dx} = e^x$$ Now, multiply both sides of the equation by \(dx\): $$du = e^x dx$$

Step 3: Rewrite the integral in terms of u

Now, rewrite the integral in terms of \(u\). Substitute \(u=e^x+1\) and \(du=e^xdx\): $$\int e^x \sec \left(e^{x}+1\right) dx =\int \sec(u) du$$

Step 4: Evaluate the integral in terms of u

We recognize that the integral in terms of \(u\) is a standard one that evaluates to the natural logarithm of the secant plus the tangent function: $$\int \sec(u) du = \ln|\sec(u) + \tan(u)| + C$$

Step 5: Substitute back x

Now, replace \(u\) with our initial substitution (\(u = e^x+1\)) to obtain the final solution: $$\ln|\sec(u) + \tan(u)| +C = \ln|\sec(e^x+1) + \tan(e^x+1)| + C$$ So, the integral of the given function is: $$\int e^{x} \sec \left(e^{x}+1\right) dx = \ln|\sec(e^x+1) + \tan(e^x+1)| + C$$

Key Concepts

Exponential Functions

Exponential functions are mathematical expressions where a constant base is raised to a variable exponent. These functions are denoted generally as \( a^x \), where \( a \) is any non-zero constant. They emerge frequently in various fields, such as biology for modeling population growth, physics for radioactive decay, and finance for compound interest. One of the most famous exponential functions is the natural exponential function \( e^x \), where \( e \) is the mathematical constant approximately equal to 2.718.

• Definition: An exponential function has the form \( f(x) = a^x \) where \( a > 0 \, \text{and} \, a eq 1 \).
• Derivative: The derivative of \( e^x \) with respect to \( x \) is \( e^x \) itself. This property makes exponential functions particularly important in calculus.
• Integration: Integrating \( e^x \) yields the result \( e^x + C \), where \( C \) is the constant of integration.

Exponential functions grow very quickly as the variable increases, and the integration of such functions often involves recognizing these growth patterns or applying substitutions to simplify the equation for easier integration.

Trigonometric Integrals

Trigonometric integrals involve the integration of trigonometric functions such as sine, cosine, tangent, and secant. These integrals are fundamental in calculus and are frequently encountered when dealing with periodic functions or analyzing waveforms. In our exercise, the integral involves the secant function.

• Standard Integrals: The integral of \( \sec(x) \) can be expressed as \( \ln|\sec(x) + \tan(x)| + C \). This result is quite standard and frequently used in calculus problems.
• Combination with Substitution: Trigonometric integrals can often be simplified using substitution techniques, especially when they are composed with other functions like exponential functions.

The secant function \( \sec(x) = 1/\cos(x) \) involves more complex trigonometric calculations compared to sine or cosine. Thus, recognizing these connections can help in formulating the basic structure of your integral solution.

Calculus Techniques

Calculus techniques such as integration by substitution are powerful methods for tackling complex integrals. This particular technique helps simplify integrals by changing variables.

• Substitution Method: The method involves selecting a portion of the integrand to replace with a new variable \( u \), making it easier to integrate. For the given integral, we chose \( u = e^x + 1 \).
• Finding \( du \): After choosing \( u \), you find \( du \) by differentiating \( u \) with respect to \( x \). Here, \( du = e^x \, dx \).
• Rewrite and Integrate: Substitute \( du \) and \( u \) back into the integral, simplifying it, then perform the integration. In this case, it simplified to \( \int \sec(u) \, du \). After integrating, substitute back the original variable.

The benefit of these techniques lies in their ability to convert challenging integrals into standard forms that are simpler to resolve. Mastery of substitution not only strengthens calculus skills but also builds a foundation for solving real-world problems efficiently.