Problem 9 - Calculus for Biology and Medicine

Question

Evaluate the indefinite integral by making the given substitution. $$ \int e^{2 x+3} d x, \text { with } u=2 x+3 $$

Step-by-Step Solution

Answer

The indefinite integral is \( \frac{1}{2} e^{2x+3} + C \).

Step 1: Identify the Substitution

In this problem, the substitution given is \( u = 2x + 3 \). This means we need to replace \( 2x + 3 \) with \( u \) in the integral.

Step 2: Differentiate the Substitution

Differentiate \( u = 2x + 3 \) with respect to \( x \) to find \( du \). The derivative is \( \frac{du}{dx} = 2 \).

Step 3: Solve for dx

Rearrange the derivative for \( dx \) as follows:\[ dx = \frac{du}{2} \]

Step 4: Substitute in the Integral

Substitute \( u \) and \( dx \) into the integral. It turns from\[ \int e^{2x+3} \, dx \] to \[ \int e^{u} \, \frac{du}{2} \] or \[ \frac{1}{2} \int e^{u} \, du \] .

Step 5: Integrate with respect to u

Integrate \( \frac{1}{2} \int e^u \, du \). The result is \[ \frac{1}{2} e^u + C \] where \( C \) is the constant of integration.

Step 6: Back-Substitute for u

Replace \( u \) with the original expression \( 2x + 3 \). The solution becomes\[ \frac{1}{2} e^{2x+3} + C \].

Key Concepts

Substitution Method

The substitution method is a powerful technique used to simplify the process of integration. It works by replacing a complex expression with a simpler variable, making the integral easier to solve. In the given exercise, the substitution is given as \( u = 2x + 3 \).

• This method often involves identifying a part of the integrand (the function inside the integral) that can be replaced by a single variable, \( u \).
• Once the substitution is chosen, you differentiate it to find \( du \). This step is essential as it helps in changing the integration variable from \( x \) to \( u \).
• In our exercise, differentiating \( u = 2x + 3 \) gives \( \frac{du}{dx} = 2 \), leading to \( dx = \frac{du}{2} \).

This transformation is crucial because it simplifies the integral to a form that is easier to integrate.

Integration

Integration is the mathematical process of finding the integral of a function. In the case of indefinite integration, the result includes a constant, \( C \), representing an infinite set of possible solutions.

• With the substitution step complete, our integral \( \int e^{2x+3} \, dx \) changes to \( \frac{1}{2} \int e^u \, du \).
• The integrand \( e^u \) is a basic exponential function, which simplifies the integration.

Once you find the antiderivative, which is \( \int e^u \, du = e^u \), the integral becomes \( \frac{1}{2} e^u + C \). This step transforms a complex expression into a simple one.

Exponential Function

Exponential functions are a key part of calculus and often appear in integration problems. These functions take the form \( e^x \), where \( e \) is the base of natural logarithms.

• They have unique properties, such as the derivative and integral of \( e^x \) both being \( e^x \), which greatly simplifies the process of calculus operations.
• In our problem, after substituting and integrating, \( e^u \) becomes the focus, and its integral is \( e^u \).

After integrating, it is important to return to the original variable by substituting \( u \) back with \( 2x + 3 \). This step gives us the final solution: \( \frac{1}{2} e^{2x+3} + C \). Understanding the behavior of exponential functions is crucial for successfully solving such integrals.