Problem 33 - Applied Calculus

Question

Find the integrals .Check your answers by differentiation. $$\int e^{3 x-4} d x$$

Step-by-Step Solution

Answer

The integral of \( e^{3x-4} \) is \( \frac{1}{3} e^{3x-4} + C \).

Step 1: Identify the Integral to be Solved

We are tasked with finding the indefinite integral of the function \( e^{3x-4} \). The integral is given as \( \int e^{3x-4} \, dx \).

Step 2: Make a Substitute to Simplify the Integral

To simplify the integration process, let's substitute \( u = 3x - 4 \). Then, the derivative of \( u \) with respect to \( x \) is \( \frac{du}{dx} = 3 \). This implies that \( dx = \frac{1}{3} du \).

Step 3: Perform the Integration

Substituting into the integral, we get:\[ \int e^u \cdot \frac{1}{3} \, du = \frac{1}{3} \int e^u \, du. \]The integral of \( e^u \) with respect to \( u \) is \( e^u \), so:\[ \frac{1}{3} e^u + C, \]where \( C \) is the constant of integration.

Step 4: Substitute Back in Terms of x

Return to the original variable by substituting \( u = 3x - 4 \) back into the expression. This gives:\[ \frac{1}{3} e^{3x-4} + C. \]

Step 5: Differentiate to Check the Solution

Differentiate the result \( \frac{1}{3} e^{3x-4} + C \) to verify that it equals the original function. The derivative with respect to \( x \) is:\[ \frac{1}{3} \cdot 3 \cdot e^{3x-4} = e^{3x-4}. \]This is the original integrand, confirming the solution is correct.

Key Concepts

Substitution Method

The substitution method is a vital tool in calculus, predominantly when dealing with integration. It simplifies an integral by substituting a part of the integrand with a new variable called 'u'.
This method is particularly useful for integrals that cannot be easily integrated by standard means. When using the substitution method, look for expressions within the integral that resemble a function and its derivative, suggesting a potential substitution. In our exercise, the integral is \( \int e^{3x-4} \, dx \).

• First, we identify \( u = 3x - 4 \).
• Calculate the derivative: \( \frac{du}{dx} = 3 \).
• Solve for \( dx \) to get \( dx = \frac{1}{3} \, du \).

This allows us to shift the integral to a simpler form: \( \int e^u \cdot \frac{1}{3} \, du \), which is much easier to handle.

Constant of Integration

The constant of integration is a core concept in indefinite integration, ensuring that no particular solution is missed. When performing indefinite integrals, we often add a constant, denoted by \( C \), because the derivative of any constant is zero. This means different functions can have the same derivative.In our example, once we integrate \( \int e^u \, du \), the result is \( e^u \).
Because we are dealing with an indefinite integral, we express the solution as \( e^u + C \).After substituting back \( u = 3x - 4 \), the solution becomes\( \frac{1}{3} e^{3x-4} + C \).
This integration constant \( C \) signifies all possible vertical shifts of the original function, retaining the generality of solutions.

Differentiation

Differentiation is the process of computing a derivative, which illustrates how a function changes. In the context of integration, differentiation serves as a check to verify our solutions.Once we have an integral solution, we can differentiate it to verify that it matches our original integrand.
In this way, we ensure that the integration was performed accurately.For our function, \( \frac{1}{3} e^{3x-4} + C \):

• Differentiate with respect to \( x \).
• The chain rule applies: \( \frac{d}{dx}[ \frac{1}{3} e^{3x-4}] = \frac{1}{3} \cdot 3 \cdot e^{3x-4} \).
• The result is \( e^{3x-4} \), which is identical to the original integrand.

This confirms the correctness of our integration, ensuring we have called on all steps and rules appropriately in our process.