Problem 52 - Applied Calculus

Question

Find the indefinite integrals. $$\int e^{-0.05 t} d t$$

Step-by-Step Solution

Answer

The integral is \(-20e^{-0.05t} + C\).

Step 1: Identify the Integral Form

The given integral has the form \(\int e^{kt} \, dt\). The constant \(k\) in this exercise is \(-0.05\).

Step 2: Apply the Exponential Integral Formula

For the integral \(\int e^{kt} \, dt\), the integration formula is \(\frac{1}{k}e^{kt} + C\), where \(C\) is the constant of integration. We apply this formula with \(k = -0.05\).

Step 3: Calculate the Integral

Substitute \(k = -0.05\) into the formula: \(\int e^{-0.05t} \, dt = \frac{1}{-0.05}e^{-0.05t} + C\). Calculating \(\frac{1}{-0.05}\), we get \(-20\).

Step 4: Express the Solution

The indefinite integral evaluated is \(-20e^{-0.05t} + C\), where \(C\) is the constant of integration.

Key Concepts

Exponential Functions

Exponential functions are a fundamental component of calculus and mathematics as a whole. These functions are in the form of \(a^x\) or \(e^{kt}\), where \(e\) is the base of the natural logarithm, approximately equal to 2.718. In our exercise, we specifically deal with \(e^{kt}\), which describes continuous growth or decay, depending on the sign of \(k\).

• If \(k\) is positive, the function represents growth.
• If \(k\) is negative, as in our exercise, it represents decay.

Exponential functions appear frequently in real-world scenarios including population growth, radioactive decay, and interest calculations.
Understanding these functions is vital because they ease the modeling of complex processes in different fields. Being comfortable with their integration is key to analyzing situations where change happens at a constantly compounding rate.

Integration Techniques

Integration is the reverse process of differentiation. The purpose of integrating a function is to find another function whose derivative is the original function. In calculus, different techniques exist to tackle various forms of integrals.
For exponential functions like \(e^{kt}\), there's a standard integration technique. The integral is computed using the formula:\[ \int e^{kt} \, dt = \frac{1}{k} e^{kt} + C \]This formula assumes that \(k\) is constant. In our example, by substituting \(k = -0.05\) into this formula, we systematically find the indefinite integral. Thus, the calculation involves:

• Determining \(k\).
• Substituting \(k\) into the formula.
• Simplifying to obtain the solution: \(-20e^{-0.05t} + C\).

These steps simplify finding indefinite integrals for exponential expressions, making them straightforward and calculable on paper or using technology.

Constant of Integration

When solving indefinite integrals, the constant of integration, \(C\), plays a crucial role. This constant emerges because when taking the antiderivative, there are infinitely many possible functions due to the property of derivative operations. Differentiating any constant results in zero, which means our integral result could originally have had any constant added to the function.
In our exercise, the solution \(-20e^{-0.05t} + C\) includes \(C\), representing all possible vertical shifts of this solution graph. Including this constant ensures we describe the most general form of the antiderivative. It's essential because:

• It acknowledges that there's not a unique solution.
• It helps apply conditions for specific solutions in practical problems, like finding particular solutions with initial conditions.

Remember, the constant of integration is not just a mere symbol but a crucial part of the solution—providing flexibility and completeness of the function you derive from the integral.