Problem 38 - Calculus and its applications

Question

Evaluate. (Be sure to check by differentiating!) $$ \int b e^{a x} d x $$

Step-by-Step Solution

Answer

The integral evaluates to \( \frac{b}{a} e^{ax} + C \).

Step 1: Identify the Integral and Its Components

The integral given is \( \int b e^{ax} \, dx \). Here, \( b \) is a constant, \( e \) is the base of the natural logarithm, and \( ax \) is the exponent.

Step 2: Recognize the Integral Formula to Use

This integral is a standard form of \( \int e^{ax} \, dx \), which evaluates to \( \frac{1}{a} e^{ax} + C \), where \( C \) is the constant of integration.

Step 3: Apply the Formula to Evaluate the Integral

Apply the constant multiple rule \( \int k \cdot f(x) \, dx = k \cdot \int f(x) \, dx \), where \( k = b \). Thus, the integral becomes \( b \cdot \int e^{ax} \, dx = b \cdot \left( \frac{1}{a} e^{ax} \right) + C = \frac{b}{a} e^{ax} + C \).

Step 4: Verify by Differentiating the Result

Differentiate \( \frac{b}{a} e^{ax} + C \) using the chain rule. The derivative \( \frac{d}{dx} \left( \frac{b}{a} e^{ax} \right) = \frac{b}{a} \cdot a \, e^{ax} = b e^{ax} \), which matches the original integrand, confirming the solution is correct.

Key Concepts

Integral Calculus

Integral Calculus is all about finding the accumulation of quantities. In simple terms, when we want to find the area under a curve, we use integrals. The integral \[\int b e^{ax} \, dx\]can represent things like growth over time or the total amount of something that changes with respect to another variable. Here:

• \(b\) is a constant that scales the function.
• \(e^{ax}\) is an exponential function indicating rapid growth or decay depending on the value of \(a\).

To solve the integral, we find an antiderivative—a function whose derivative is the original function inside the integral. We add \(C\), the constant of integration, because integration can shift up or down along the y-axis without changing its derivative.
For more complex functions, we have special techniques such as substitution and integration by parts, often necessary for non-standard integrals.

Exponential Functions

An exponential function involves the constant \(e\), approximately 2.71828, raised to a variable power. In mathematics and especially in calculus, exponential functions model continuous growth or decay. When dealing with the exponential function \(e^{ax}\), different values of \(a\) can represent different rates of change.

• If \(a > 0\), \(e^{ax}\) describes rapid growth.
• If \(a < 0\), \(e^{ax}\) represents decay.

A key property of exponential functions is that their rate of growth is proportional to their value, represented mathematically by:
The derivative of the function \(e^{ax}\) with respect to \(x\) is \(a e^{ax}\). This property gives exponential functions their unique capability to model real-world processes such as population growth, radioactive decay, and compound interest.

Differentiation Verification

Verification by differentiation ensures that our calculated integral is correct. Once you find an antiderivative, you take its derivative to see if you get the original function back. This process acts as a double-check for our integral calculations.

• Start with the antiderivative, \(\frac{b}{a} e^{ax} + C\).
• Differentiate it using the chain rule: the derivative of \(e^{ax}\) is \(a e^{ax}\).
• Multiply \(a\) by the constant \(\frac{b}{a}\) to get \(b e^{ax}\), which should mirror our original integrand.

If the derivative matches the initial function inside the integral, this confirms the evaluated integral is correct. Differentiation verification is common practice in calculus to ensure accuracy and deepen understanding of the relationship between functions and their rates of change.