Problem 62 - Applied Calculus

Question

Find the indefinite integrals. $$\int e^{2 t} d t$$

Step-by-Step Solution

Answer

\( \int e^{2t} \, dt = \frac{1}{2}e^{2t} + C \)

Step 1: Identify the Integral Form

We are given the integral \( \int e^{2t} \, dt \). Recognize that this is of the form \( \int e^{kt} \, dt \) where \( k = 2 \). The integral of \( e^{kt} \) is generally \( \frac{1}{k}e^{kt} + C \) where \( C \) is the constant of integration.

Step 2: Apply the Integral Formula

Using the identified form, apply the formula for integration: \( \int e^{2t} \, dt = \frac{1}{2}e^{2t} + C \). Here we have divided by the constant \( k = 2 \) which is the coefficient of \( t \) in the exponent.

Step 3: Simplify the Expression (if needed)

While there is no complex simplification needed beyond applying the formula, ensure the integration constant \( C \) is included in the final answer as it represents the family of antiderivatives.

Key Concepts

Exponential Functions

Exponential functions are crucial in calculus because they describe phenomena where growth or decay happens at a constant rate per unit of time. The standard form is expressed as \( e^{kt} \), where \( e \) is the base of the natural logarithm (approximately 2.718) and \( k \) is a constant that affects the rate of growth or decay.

• If \( k > 0 \), the function represents exponential growth.
• If \( k < 0 \), the function represents exponential decay.

What makes exponential functions particularly interesting in the context of calculus is their unique property: when you differentiate or integrate an exponential function of base \( e \), it results in another exponential function. This property simplifies the process of finding derivatives and integrals, making these functions a favorite in both theoretical and applied mathematics problems.

Integration Formula

The integration formula for exponential functions allows us to find their indefinite integrals easily. An indefinite integral is essentially the inverse operation to differentiation, and it's used to find a function whose derivative is the given function. In our problem, we're dealing with the indefinite integral of an exponential function: \( \int e^{kt} \, dt \).
The integration formula is:\( \int e^{kt} \, dt = \frac{1}{k}e^{kt} + C \)
This formula tells us:

• We divide by \( k \), the coefficient of \( t \) in the exponent, to adjust for the rate of change in the exponential function.
• We add \( C \), the constant of integration, which represents all the possible vertical shifts of the function. More on that in the next section.

In our specific exercise, \( k = 2 \), leading us directly to:\(\int e^{2t} \, dt = \frac{1}{2}e^{2t}+ C \).
This straightforward application of the integration formula lets us seamlessly integrate exponential functions.

Constant of Integration

In the process of finding indefinite integrals, the constant of integration \( C \) plays a vital role. It is not just an arbitrary number thrown in at the end of solving integrals. Rather, it accounts for all possible antiderivatives of the given function. When differentiating, the constant \( C \) disappears, but upon integration, we must reintroduce this constant to represent every possible vertical shift of the antiderivative that results in the original function.
The constant of integration is important because:

• The indefinite integral does not specify one unique antiderivative; instead, it describes a family of functions that differ by a constant.
• Each different value of \( C \) corresponds to a different function within this family.

For instance, the solutions \( \int e^{2t} \, dt = \frac{1}{2}e^{2t} + 0 \), \( \frac{1}{2}e^{2t} + 1 \), and \( \frac{1}{2}e^{2t} - 5 \) all have the same derivative \( e^{2t} \). Hence, the constant \( C \) ensures that the integral represents an entire set of valid solutions.