Problem 60 - Calculus for Biology and Medicine

Question

Compute the indefinite integrals. $$ \int 2 e^{-x / 3} d x $$

Step-by-Step Solution

Answer

The indefinite integral is \(-6 e^{-x/3} + C\), where \( C \) is the constant of integration.

Step 1: Identify the Integration Formula

The integral we need to solve is of the form \( \int a e^{bx} dx \). We know that the integral of an exponential function of the form \( \int e^{ax} dx \) is \( \frac{1}{a} e^{ax} + C \), where \( C \) is the constant of integration.

Step 2: Identify Constants a and b

The function given to us is \( 2 e^{-x/3} \). Here, \( a = 2 \) and \( b = -1/3 \). This means the integral formula can be applied to \( e^{(-1/3)x} \).

Step 3: Apply the Integration Formula

Using \( b = -1/3 \), for the integral \( \int e^{(-1/3)x} dx \), we use the formula \( \int e^{bx} dx = \frac{1}{b} e^{bx} + C \). Applying this, we get \( \frac{1}{-1/3} e^{-x/3} + C = -3 e^{-x/3} + C \).

Step 4: Include the Constant Multiplier

Since there is a constant \( a = 2 \) multiplying the exponential, we multiply our result by this constant. The final integral result is \( 2 \times (-3) e^{-x/3} + C = -6 e^{-x/3} + C \).

Step 5: State the Final Indefinite Integral

Thus, the indefinite integral of the given function is \(-6 e^{-x/3} + C\), where \( C \) is the integration constant.

Key Concepts

Integration Formula

When tackling indefinite integrals, understanding the integration formula is crucial. An integral represents the area under a function and is the reverse operation of differentiation. To solve an integral like \( \int a e^{bx} dx \), you'll first want to recall the formula for exponential functions. The basic integration formula for exponential functions is:

• \( \int e^{ax} dx = \frac{1}{a} e^{ax} + C \)

This formula tells us that the integral of an exponential function is a scaled version of itself. The scaling factor is the reciprocal of the coefficient of \( x \) in the exponent.
Bearing this in mind, the process becomes straightforward by substituting the constants into the formula. This makes dealing with complex exponential integrals much more manageable.

Exponential Function

Exponential functions are essential components in calculus and can appear in a variety of forms. They are characterized by the constant base raised to a power that is a linear function of the variable, such as \( e^{bx} \). Here, the constant \( e \) is a mathematical number approximately equal to 2.71828, known as Euler's number.
One of the key properties of exponential functions is that they exhibit continuous growth or decay, described by the exponent. In the context of integration, the nature of the exponent dictates the behavior of the integral. For example, when integrating \( e^{(-1/3)x} \), the exponent impacts how we scale the function during integration.

• Growth occurs when \( b \) is positive.
• Decay occurs when \( b \) is negative.

Understanding these properties allows us to manipulate the function correctly during integration.

Constant of Integration

The constant of integration, represented by \( C \), is a fundamental part of indefinite integrals in calculus. Since integration is the reverse process of differentiation, determining the original function from its derivative can contain an infinite number of constants that disappeared when differentiating.
This is where the constant of integration comes into play. It accounts for all the possible vertical shifts of the function, ensuring that every potential original function is captured. Without this constant, an indefinite integral would only represent one possible solution instead of the entire family of solutions.
For any indefinite integral calculation, always include \( C \) at the end to signify the general solution. For instance, any time you solve an integral like \( \int 2 e^{-x/3} dx \), don't forget to append \( + C \) to your result, reflecting this inherent uncertainty in the original function's vertical starting point.