Evaluating The Integral Of 5e^(3x) Dx A Step-by-Step Guide
In this comprehensive guide, we will delve into the step-by-step process of evaluating the integral of 5e^(3x) dx. This type of integral is commonly encountered in calculus and has applications in various fields such as physics, engineering, and economics. Understanding how to solve this integral is crucial for anyone studying or working in these areas. We will break down the problem into manageable steps, explaining the underlying principles and techniques involved. Whether you're a student learning calculus or a professional needing a refresher, this guide will provide you with a clear and thorough understanding of how to evaluate this integral.
Understanding the Basics of Integration
Before we dive into the specifics of evaluating the integral of 5e^(3x) dx, it's essential to have a solid grasp of the basics of integration. Integration, in simple terms, is the reverse process of differentiation. While differentiation helps us find the rate of change of a function, integration helps us find the original function given its rate of change. The integral symbol, ∫, represents the process of integration, and the expression following it is the integrand, which in our case is 5e^(3x). The 'dx' at the end signifies that we are integrating with respect to the variable x. The result of an integration is a function, often denoted by a capital letter, plus a constant of integration, denoted by 'C'. This constant arises because the derivative of a constant is always zero, so when we reverse the process, we need to account for any possible constant term. The power rule, the constant multiple rule, and the sum/difference rule are the basis of integration.
The power rule states that the integral of x^n with respect to x is (x^(n+1))/(n+1) + C, where n is any real number except -1. The constant multiple rule states that the integral of a constant times a function is equal to the constant times the integral of the function. The sum/difference rule states that the integral of a sum or difference of functions is equal to the sum or difference of their integrals, respectively. These basic rules, along with the knowledge of derivatives of common functions, form the foundation for solving more complex integrals. In the case of exponential functions like e^(3x), we need to recall that the derivative of e^x is e^x itself, and this knowledge will be crucial in evaluating our integral.
Step 1: Applying the Constant Multiple Rule
The first step in evaluating the integral of 5e^(3x) dx is to apply the constant multiple rule. This rule states that the integral of a constant times a function is equal to the constant times the integral of the function. In mathematical terms, ∫kf(x) dx = k∫f(x) dx, where k is a constant and f(x) is a function. In our case, the constant is 5 and the function is e^(3x). Applying the constant multiple rule, we can rewrite the integral as follows: ∫ 5e^(3x) dx = 5∫ e^(3x) dx. This step simplifies the problem by isolating the exponential function, which is easier to integrate. The constant multiple rule is a powerful tool in integration as it allows us to focus on integrating the essential part of the function without the distraction of constant coefficients. By applying this rule, we have effectively reduced our problem to finding the integral of e^(3x) dx, which is a more manageable task. The next step will involve using a technique called u-substitution to further simplify the integral.
Step 2: Using u-Substitution to Simplify the Integral
To evaluate the integral of e^(3x) dx, we employ a technique called u-substitution. U-substitution is a powerful method used to simplify integrals by replacing a complex expression with a single variable, making the integral easier to solve. The key to successful u-substitution is choosing the right expression to substitute. In this case, we let u = 3x. This substitution is motivated by the fact that the derivative of 3x is a constant, which will help simplify the exponential term. Now, we need to find the differential of u, denoted as du. Taking the derivative of u with respect to x, we get du/dx = 3. Multiplying both sides by dx, we obtain du = 3dx. However, our integral contains dx, not 3dx. To resolve this, we divide both sides of the equation by 3, giving us dx = (1/3)du. Now we can substitute both u and dx into our integral. Replacing 3x with u and dx with (1/3)du, the integral 5∫ e^(3x) dx becomes 5∫ e^u (1/3)du. This substitution has transformed the integral into a simpler form involving the exponential function e^u and a constant factor. The next step is to apply the constant multiple rule again to further simplify the integral.
Step 3: Applying the Constant Multiple Rule Again
After performing the u-substitution, our integral now looks like 5∫ e^u (1/3)du. We can further simplify this by applying the constant multiple rule again. The constant multiple rule allows us to move the constant (1/3) outside the integral sign. So, we can rewrite the integral as 5 * (1/3) ∫ e^u du. Multiplying the constants 5 and (1/3) gives us 5/3. Therefore, our integral now becomes (5/3) ∫ e^u du. This step has simplified the integral to a point where it is easily recognizable and solvable. The integral of e^u with respect to u is a fundamental integral that we should know. It is a direct application of the fact that the derivative of e^u is e^u, so the integral of e^u is also e^u. With this knowledge, we can proceed to the next step, which is evaluating the integral of e^u du.
Step 4: Integrating e^u with Respect to u
Now that we have simplified our integral to (5/3) ∫ e^u du, we can proceed with the integration. The integral of e^u with respect to u is a fundamental result in calculus. Recall that the derivative of e^u is e^u. Therefore, the integral of e^u is also e^u, plus a constant of integration, denoted as C. So, ∫ e^u du = e^u + C. Applying this result to our integral, we get (5/3) ∫ e^u du = (5/3)(e^u + C). It's important to remember the constant of integration, C, because the derivative of a constant is zero, so there could be any constant term in the original function. However, for now, we will keep the constant C separate and address it in the final step. We have successfully integrated the exponential function e^u with respect to u. The next step is to substitute back the original variable x to express our result in terms of x.
Step 5: Substituting Back to the Original Variable
We have now evaluated the integral in terms of u, but our original problem was in terms of x. Therefore, we need to substitute back our original expression for u, which was u = 3x. Replacing u with 3x in our result (5/3)e^u, we get (5/3)e^(3x). This substitution is crucial because it expresses the result in terms of the original variable, which is what we ultimately want. We are now very close to the final answer. We have performed the integration and substituted back the original variable. The only remaining step is to add the constant of integration, C, and simplify the expression if necessary. This final step will give us the complete and correct answer to the integral of 5e^(3x) dx. The constant of integration is a crucial part of indefinite integrals, as it represents the family of functions that have the same derivative. In the next step, we will add the constant of integration and present the final result.
Step 6: Adding the Constant of Integration and Final Result
In the previous steps, we found that the integral of 5e^(3x) dx is (5/3)e^(3x). However, we must not forget to add the constant of integration, C. The constant of integration is added because the derivative of a constant is always zero, so when we reverse the process of differentiation (i.e., integration), we need to account for any possible constant term that might have been present in the original function. Therefore, the complete result is (5/3)e^(3x) + C. This is the final answer to our problem. We have successfully evaluated the integral of 5e^(3x) dx by applying the constant multiple rule, u-substitution, and the fundamental integral of e^u. The constant of integration, C, represents an arbitrary constant and indicates that there are infinitely many functions that have the same derivative. To summarize, the integral of 5e^(3x) dx is (5/3)e^(3x) + C, where C is the constant of integration. This result is a fundamental example of how to integrate exponential functions and demonstrates the power of u-substitution in simplifying integrals.
Conclusion
In conclusion, we have successfully evaluated the integral of 5e^(3x) dx using a combination of techniques, including the constant multiple rule and u-substitution. The step-by-step approach we followed allowed us to break down the problem into manageable parts, making it easier to understand and solve. We started by applying the constant multiple rule to separate the constant 5 from the exponential function. Then, we used u-substitution to simplify the integral by letting u = 3x. This substitution transformed the integral into a simpler form involving e^u. We then integrated e^u with respect to u, which gave us e^u + C. Finally, we substituted back the original variable x and added the constant of integration, C, to obtain the final result: (5/3)e^(3x) + C. This example demonstrates the importance of understanding fundamental integration techniques and how to apply them effectively. The skills learned in this guide can be applied to a wide range of integration problems, making it a valuable resource for students and professionals alike. Mastering integration techniques is essential for success in calculus and related fields.