Problem 31 - Calculus and its applications

Question

Find each integral. $$ \int 5 e^{3 x} d x $$

Step-by-Step Solution

Answer

The integral is \( \frac{5}{3} e^{3x} + C_1 \).

Step 1: Identify the Integral Form

Notice that the integral has the form \( \int C e^{kx} \, dx \), where \( C = 5 \) and \( k = 3 \). The integral of \( e^{kx} \) is \( \frac{1}{k} e^{kx} \). So in this case, we are dealing with an integral that fits this formula.

Step 2: Apply the Integral Formula

Using the formula \( \int C e^{kx} \, dx = \frac{C}{k} e^{kx} + C_1 \), we can substitute \( C = 5 \) and \( k = 3 \) into it. This gives \( \int 5 e^{3x} \, dx = \frac{5}{3} e^{3x} + C_1 \).

Step 3: Add the Constant of Integration

Remember to include the constant of integration, \( C_1 \), because we are dealing with an indefinite integral. This constant accounts for any constant term that could have been differentiated to zero in the integration process.

Key Concepts

Indefinite Integral

An indefinite integral is an integral without specified limits of integration. It refers to a function that results from integrating another function, and it generally represents a family of functions. When you compute an indefinite integral, you're essentially

• Trying to find the original function whose derivative is known.
• Its result is a general formula plus a constant of integration.

The indefinite integral is denoted by the integral symbol followed by the function and its differential, like this: \( \int f(x)\, dx \). This process is the reverse of differentiation.
In the given problem, we have the indefinite integral \( \int 5 e^{3x} \, dx \). We want to find the function that, when differentiated, gives us \( 5 e^{3x} \). By applying the formula for integrating exponential functions, we can discover this family of solutions.

Constant of Integration

In calculus, the constant of integration is crucial when working with indefinite integrals. When you integrate a function, you're finding an entire family of possible antiderivatives or original functions. This is because differentiation a constant results in zero.
As a result, adding a constant doesn't change the derived function. Therefore, when integrating, we add a term \( C \) to account for all these possibilities. This \( C \) is called the constant of integration.
In our original exercise, we added the constant of integration \( C_1 \) to the result of \( \int 5 e^{3x} \, dx \) which gives us \[ \frac{5}{3} e^{3x} + C_1 \] This constant reminds us that there are multiple functions that could differentiate to give \( 5 e^{3x} \).
Whenever you encounter an indefinite integral, don't forget to include the constant of integration to complete your solution.

Integral Formula

Understanding integral formulas can help you tackle complex integrals with ease. In particular, for exponential functions of the form \( e^{kx} \), the integral formula is fundamental. The general formula for the integral of \( e^{kx} \) is:\[ \int e^{kx} \, dx = \frac{1}{k} e^{kx} + C \]This indicates that when you integrate \( e^{kx} \), the result is a fraction of the same exponential function, with the fraction's denominator being proportional to the constant \( k \) from the exponent.
Applying this concept to our problem, we integrated \( 5 e^{3x} \) using:

• Identifying \( C = 5 \) and \( k = 3 \).
• Substituting these values into the formula.

This gave us:\[ \int 5 e^{3x} \, dx = \frac{5}{3} e^{3x} + C_1 \]By using this formula, we were able to solve the exercise efficiently.