Problem 65 - Applied Calculus

Question

Find a formula for \(\int e^{a x+b} d x\) where \(a\) and \(b\) are constants.

Step-by-Step Solution

Answer

The formula for \( \int e^{ax+b} \, dx \) is \( \frac{1}{a} e^{ax+b} + C \).

Step 1: Recognize the integral form

The given integral is \( \int e^{ax+b} \, dx \). This integral involves an exponential function which can be solved using substitution.

Step 2: Choose a suitable substitution

To simplify the integral, substitute \( u = ax + b \). This substitution helps to transform the function into a simpler form involving \( u \).

Step 3: Compute the differential

Differentiate \( u = ax + b \) to find \( du \). Therefore, \( \frac{du}{dx} = a \), which implies \( du = a \, dx \). Thus, \( dx = \frac{du}{a} \).

Step 4: Apply the substitution

Substitute \( u = ax + b \) and \( dx = \frac{du}{a} \) into the integral. This gives: \[ \int e^{ax+b} \, dx = \int e^u \cdot \frac{du}{a} = \frac{1}{a} \int e^u \, du. \]

Step 5: Integrate the exponential function

The integral of \( e^u \) with respect to \( u \) is \( e^u \). So, we have: \[ \frac{1}{a} \int e^u \, du = \frac{1}{a} e^u + C, \] where \( C \) is the constant of integration.

Step 6: Substitute back in terms of x

Replace \( u \) with \( ax+b \) to express the antiderivative in terms of \( x \). Thus, the antiderivative becomes: \[ \frac{1}{a} e^{ax+b} + C. \]

Key Concepts

Exponential Function

The exponential function is a mathematical function often written as \( e^x \), where \( e \) is the Euler's number approximately equal to 2.71828. It is a unique and powerful function used widely in calculus, as it has the derivative property such that the derivative of \( e^x \) is itself, \( e^x \). This property makes it especially useful for solving differential equations and working within the realm of calculus.
In the given exercise, the function involves an exponential term, \( e^{ax+b} \), where \( a \) and \( b \) are constants. This means the input to the exponential, known as the exponent, is a linear expression involving \( x \). This setup is common in calculus problems, especially those that require integration or differentiation of exponential functions.

• The exponential function grows at an increasing rate, meaning as \( x \) values increase, \( e^x \) becomes extremely large very quickly.
• It is continuous and differentiable for all real numbers, which helps in finding solutions to many calculus problems.

Understanding the exponential function is crucial for tackling problems involving growth processes, decay processes, and in scenarios where rapid changes occur.

Substitution Method

The substitution method is a popular technique in calculus used to simplify the process of integrating complex functions. In this method, you choose a new variable, usually denoted as \( u \), to replace an expression in the original integral.
This technique was applied in the given exercise by choosing \( u = ax + b \). The aim of substitution is two-fold: it transforms the integral into a simpler form and it also makes calculation easier by reducing it to a standard integral that we know how to solve.

• First, identify the inner function. This is the expression you will substitute, like \( ax + b \) in our example.
• Calculate \( du \), the derivative of \( u \), and rearrange it in terms of \( dx \). This helps when you replace \( dx \) in the original integral with \( \frac{du}{a} \).

Substitution essentially "repackages" the integral, allowing for an easier path to finding the antiderivative. It's a crucial skill for handling various integration challenges, particularly when faced with compound or composite functions.

Antiderivative

The antiderivative, also known as the integral, is a fundamental concept in calculus that represents the reverse process of differentiation. Finding an antiderivative involves determining a function whose derivative gives the original function.
In our exercise, after substitution we obtained the integral \( \int e^u \, du \), which is a straightforward one because we know that the antiderivative of \( e^u \) is simply \( e^u \). Therefore, for this integral, the antiderivative is \( \frac{1}{a} e^u + C \), where \( C \) is a constant.

• The power of understanding antiderivatives lies in solving differential equations and modeling situations that involve areas under curves.
• Remember, every function can have infinitely many antiderivatives, differing by an additive constant.

Finding the right antiderivative can solve many practical problems like computing displacement, computing areas, and predicting systems behavior over time.

Constant of Integration

The constant of integration, often denoted by \( C \), is an important element in indefinite integration. When you calculate the antiderivative of a function, the result includes \( C \) because integration, unlike differentiation, is not a unique process.
The lack of uniqueness arises because the derivative of any constant is zero. Therefore, any constant added to an antiderivative will still have the same derivative. For example, integrating \( e^{ax+b} \) results in \( \frac{1}{a} e^{ax+b} + C \). Adding this constant makes the solution comprehensive:

• It signifies unknown boundaries in indefinite integrals.
• Each value of \( C \) corresponds to a family of curves, all of which are identical except for their vertical position on a graph.

In practical applications, \( C \) is determined by initial conditions or boundary values of the problem to pinpoint the particular solution needed. This determination is crucial when transitioning from general integrals to applying them in real-world scenarios.