Calculating The Integral Of 2x - 5x^2 - 5x + 6 A Step-by-Step Guide
This article delves into the process of determining the integral of the function 2x - 5x^2 - 5x + 6 with respect to x. We'll break down the steps involved, explain the underlying principles of integral calculus, and provide a clear, concise solution. Understanding integration is crucial in various fields, including physics, engineering, and economics, as it allows us to calculate areas, volumes, and other important quantities.
Understanding the Basics of Integration
Before we dive into the specific problem, let's establish a firm grasp of the fundamentals of integration. Integration, at its core, is the reverse process of differentiation. While differentiation finds the rate of change of a function, integration finds the original function given its rate of change. In simpler terms, if you differentiate a function and then integrate the result, you should ideally arrive back at the original function (with the potential addition of a constant, which we'll discuss later).
The process of integration can be visualized as finding the area under a curve. Imagine plotting the function 2x - 5x^2 - 5x + 6 on a graph. The integral of this function between two points on the x-axis represents the area bounded by the curve, the x-axis, and the vertical lines at those two points. This geometrical interpretation makes integration a powerful tool for solving problems involving areas and volumes.
The symbol for integration is ∫, often referred to as the integral sign. The expression ∫f(x) dx signifies the integral of the function f(x) with respect to the variable x. The 'dx' indicates that we are integrating with respect to x, which means we are considering how the function changes as x changes. It's crucial to include 'dx' in the integral notation as it specifies the variable of integration and ensures the correct application of integration rules.
The result of integration is not a single function but a family of functions. This is because the derivative of a constant is always zero. Therefore, when we reverse the differentiation process, we lose information about any constant term that might have been present in the original function. To account for this, we add a constant of integration, denoted by 'C', to the result of every indefinite integral. This constant represents an arbitrary vertical shift of the function, meaning any function of the form F(x) + C will have the same derivative, where F(x) is the antiderivative of f(x).
Applying the Power Rule of Integration
Now, let's focus on the specific function at hand: 2x - 5x^2 - 5x + 6. To find its integral, we'll primarily use the power rule of integration. The power rule is a fundamental rule in calculus that simplifies the integration of power functions, which are functions of the form x^n, where n is any real number except -1. The power rule states that the integral of x^n with respect to x is (x^(n+1))/(n+1) + C, where C is the constant of integration.
This rule is derived from the reverse of the power rule of differentiation. When we differentiate x^n, we multiply by n and reduce the exponent by 1, resulting in nx^(n-1). Integration, being the reverse process, involves increasing the exponent by 1 and dividing by the new exponent. The power rule is widely applicable and forms the basis for integrating many polynomial functions.
To apply the power rule effectively, we need to consider each term of the function 2x - 5x^2 - 5x + 6 separately. We can break down the integral of the entire function into the sum of the integrals of its individual terms. This is because the integral of a sum (or difference) is equal to the sum (or difference) of the integrals. This property of linearity simplifies the integration process significantly.
For instance, to integrate 2x, we recognize that it's in the form of a power function where the exponent of x is 1. Applying the power rule, we increase the exponent by 1 to get 2 and divide by the new exponent, resulting in (2x^2)/2, which simplifies to x^2. Similarly, for the term -5x^2, we increase the exponent by 1 to get 3 and divide by the new exponent, leading to (-5x^3)/3. This process is repeated for each term in the function.
Remember to always include the constant of integration, C, after performing the integration. This constant represents the family of functions that have the same derivative. Omitting C would mean we're only finding one specific antiderivative, not the general antiderivative.
Step-by-Step Solution for Integrating 2x - 5x^2 - 5x + 6
Let's now systematically determine the integral of the given function, 2x - 5x^2 - 5x + 6, with respect to x. We'll break down the process into manageable steps, applying the principles and rules we've discussed earlier.
Step 1: Rewrite the function (if necessary)
In this case, the function 2x - 5x^2 - 5x + 6 is already in a suitable form for integration. However, it's helpful to combine like terms to simplify the expression. We can combine the 2x and -5x terms, resulting in * -3x*. So, the function becomes * -5x^2 - 3x + 6*.
Step 2: Apply the linearity of integration
The linearity of integration allows us to integrate each term separately. This means we can write the integral of the function as the sum of the integrals of its individual terms:
∫(-5x^2 - 3x + 6) dx = ∫(-5x^2) dx + ∫(-3x) dx + ∫(6) dx
Step 3: Apply the power rule to each term
Now, we apply the power rule to each individual term:
- ∫(-5x^2) dx = -5 * ∫(x^2) dx = -5 * (x^(2+1))/(2+1) = (-5x^3)/3
- ∫(-3x) dx = -3 * ∫(x) dx = -3 * (x^(1+1))/(1+1) = (-3x^2)/2
- ∫(6) dx = 6 * ∫(x^0) dx = 6 * (x^(0+1))/(0+1) = 6x
Note that we treated 6 as 6*x^0 to apply the power rule since any number raised to the power of 0 is 1.
Step 4: Combine the results and add the constant of integration
Finally, we combine the results from Step 3 and add the constant of integration, C:
∫(-5x^2 - 3x + 6) dx = (-5x^3)/3 - (3x^2)/2 + 6x + C
Therefore, the integral of the function 2x - 5x^2 - 5x + 6 with respect to x is (-5x^3)/3 - (3x^2)/2 + 6x + C.
Verifying the Solution
To ensure our solution is correct, we can differentiate the result we obtained and check if it matches the original function. Differentiating (-5x^3)/3 - (3x^2)/2 + 6x + C with respect to x, we get:
- d/dx [(-5x^3)/3] = -5x^2
- d/dx [(-3x^2)/2] = -3x
- d/dx [6x] = 6
- d/dx [C] = 0
Adding these results together, we get * -5x^2 - 3x + 6*, which is equivalent to our original function 2x - 5x^2 - 5x + 6 after combining like terms. This verification step confirms that our integration process was performed correctly.
Conclusion: Mastering Integral Calculus
In conclusion, we have successfully determined the integral of the function 2x - 5x^2 - 5x + 6 with respect to x. We achieved this by understanding the fundamentals of integration, applying the power rule, and utilizing the linearity of integration. The result, (-5x^3)/3 - (3x^2)/2 + 6x + C, represents the family of functions whose derivative is the original function.
The key takeaways from this exercise include the importance of the power rule in integrating polynomial functions, the significance of the constant of integration, and the ability to verify the solution through differentiation. Mastering integral calculus is a crucial skill for anyone pursuing studies in mathematics, science, or engineering, as it provides a powerful tool for solving a wide range of problems.
By consistently practicing and applying these principles, you can develop a strong foundation in integral calculus and confidently tackle more complex integration problems. Remember, practice is key to mastering any mathematical concept, and integration is no exception. The more you work through various examples, the more comfortable and proficient you will become in applying these techniques.
