Hey guys! Let's dive into the fascinating world of calculus and learn how to integrate the expression 5e^x * 13 * 3 * 4x dx. Don't worry, it might seem a bit daunting at first, but with a clear understanding of the rules and a methodical approach, we'll break it down into manageable steps. This guide is designed to make the integration process as clear as possible, even if you're just starting out. We'll explore each step with explanations, ensuring that you grasp not just how to solve it, but why we do it a certain way. By the end of this tutorial, you'll be able to confidently tackle similar integration problems. Let's get started!

Understanding the Basics of Integration

Before we begin, let's quickly recap some fundamental concepts that are crucial for understanding integration. Integration, in its simplest form, is the reverse process of differentiation. When you differentiate a function, you find its rate of change. Integration, on the other hand, finds the area under the curve of a function. The main objective of the calculation is to find the indefinite integral of the expression, and we are going to use the integration rules. The indefinite integral represents a family of functions, as the constant term vanishes when taking the derivative. When performing indefinite integration, we must not forget to include the integration constant, often denoted as C. This constant arises because the derivative of a constant is always zero. This means that when we take the anti-derivative, there's always an unknown constant term that could have been present in the original function. The rules such as the power rule, constant multiple rule, and sum/difference rule will be useful. The power rule involves increasing the power of the variable by one and dividing by the new power. The constant multiple rule states that constants can be moved outside the integral sign. The sum/difference rule allows us to integrate terms separately. Also, it's very important to become familiar with basic integration formulas, for instance, we should know that the integral of e^x is e^x plus a constant C. Understanding these basic principles is the foundation upon which the more complex integration problems, such as the one we're dealing with here, are built. Having these basics solidifies your understanding and makes the problem-solving process much smoother.

Essential Integration Rules to Remember

• The Power Rule: ∫xⁿ dx = (x^(n+1))/(n+1) + C, where n ≠ -1.
• Constant Multiple Rule: ∫cf(x) dx = c∫f(x) dx, where c is a constant.
• Sum/Difference Rule: ∫[f(x) ± g(x)] dx = ∫f(x) dx ± ∫g(x) dx.
• Integral of e^x: ∫e^x dx = e^x + C

Step-by-Step Integration of 5e^x * 13 * 3 * 4x dx

Alright, let's get into the meat of the matter! We'll break down the integration of 5e^x * 13 * 3 * 4x dx into easily digestible steps. This is a multi-step process, so be sure to follow along carefully. The main strategy is to simplify the expression first and use the integration rules. Ready?

Step 1: Simplify the Expression

First, let's simplify the constant terms in the expression. We have 5, 13, 3, and 4. Multiply these constants together: 5 * 13 * 3 * 4 = 780. So, our expression becomes: 780xe^x dx. This is much cleaner and easier to work with. Always look for opportunities to simplify your expressions before proceeding with more complex calculations. This simplification step makes the subsequent integration steps much easier and reduces the chances of making a mistake. The original expression can be simplified to a product of a constant and a function of x, which can then be integrated using the product rule of integration.

Step 2: Apply Integration by Parts

Now, we'll use a technique called integration by parts. This is necessary because our expression is a product of two functions of x: x and e^x. The integration by parts formula is derived from the product rule of differentiation and states: ∫u dv = uv - ∫v du. To apply this, we need to choose 'u' and 'dv'. A good rule of thumb is to use the LIATE rule, which is the order of preference to choose u: Logarithmic, Inverse trigonometric, Algebraic, Trigonometric, Exponential. In our case, the LIATE rule suggests that we choose u = x (algebraic) and dv = e^x dx (exponential). Therefore:

• u = x => du = dx
• dv = e^x dx => v = ∫e^x dx = e^x

Step 3: Apply the Integration by Parts Formula

Now that we've identified u, v, du, and dv, we can substitute them into the integration by parts formula: ∫u dv = uv - ∫v du. Remember that our original expression has a constant multiplier of 780, so we apply this to the final result. Substituting our values gives us:

∫780xe^x dx = 780(xe^x - ∫e^x dx)

Step 4: Integrate the Remaining Term

We're almost there, guys! We need to integrate the remaining term, which is ∫e^x dx. As we know, the integral of e^x is simply e^x. Therefore, ∫e^x dx = e^x + C. Plugging this back into our equation, we get:

780(xe^x - e^x) + C

Step 5: Final Result

And we have the final result! The integral of 5e^x * 13 * 3 * 4x dx is 780xe^x - 780e^x + C, which can be further factored to 780e^x(x - 1) + C. This is the indefinite integral, which represents a family of functions. The constant of integration, C, is essential because the derivative of any constant is zero. Therefore, when we find an anti-derivative, we must include C to account for any possible constant term in the original function. Make sure to keep this C in your final answer.

Practical Tips and Tricks

• Practice Regularly: The more you practice, the more comfortable you'll become with integration. Work through various examples to reinforce your understanding.
• Understand the Rules: Make sure you thoroughly understand the fundamental rules of integration. This will make problem-solving much easier.
• Simplify First: Always try to simplify the expression before starting the integration process. This can often make the problem much less complicated.
• Check Your Answer: After solving an integration problem, differentiate your result to see if you get back the original expression. This is a great way to check your work.

Conclusion: Mastering the Integration of 5e^x * 13 * 3 * 4x dx

Congratulations, we have successfully integrated the expression 5e^x * 13 * 3 * 4x dx! By following the steps outlined, you've seen how to break down a complex integration problem into manageable pieces. Remember that the key is to apply the right rules, simplify where possible, and take your time. This problem showed us how integration by parts, simplification, and a solid understanding of fundamental rules could come together to solve a complex problem. Keep practicing, and you'll find that integration becomes more intuitive with each problem you solve. Mastering integration requires practice and a systematic approach. This guide should provide you with the tools you need to succeed. Feel free to revisit this guide whenever you need a refresher.

Further Exploration

• Different Integration Techniques: Explore other integration techniques, such as u-substitution. This method is useful for integrating composite functions. This expands your toolkit for solving various problems. Mastering multiple integration techniques can greatly enhance your problem-solving capabilities.
• Definite Integrals: Once you're comfortable with indefinite integrals, move on to definite integrals. These have limits and give you a numerical answer representing the area under a curve between two points.
• Integration Software: Use online integration tools to check your answers and see how the problems are solved. These tools can be very helpful for learning and understanding the process.

Keep up the great work, and happy integrating!