Evaluate The Integral. 5 X2 + 25 Dx 0

Evaluating the Definite Integral: ∫₀⁵ (5x² + 25) dx

This article provides a comprehensive guide on how to evaluate the definite integral ∫₀⁵ (5x² + 25) dx. We'll break down the process step-by-step, explaining the fundamental concepts of integration and demonstrating the techniques involved. We'll also explore the broader context of definite integrals and their applications in various fields.

Understanding Definite Integrals

A definite integral is a mathematical concept that represents the signed area between a curve and the x-axis over a specified interval. Unlike indefinite integrals, which result in a family of functions, definite integrals yield a numerical value. The notation for a definite integral is:

∫ₐᵇ f(x) dx

where:

• a is the lower limit of integration.
• b is the upper limit of integration.
• f(x) is the integrand, the function being integrated.
• dx indicates that the integration is with respect to the variable x.

The definite integral is calculated by finding the antiderivative (also known as the indefinite integral) of the function f(x), evaluating it at the upper limit (b), and then subtracting the value of the antiderivative evaluated at the lower limit (a). This is expressed by the Fundamental Theorem of Calculus:

∫ₐᵇ f(x) dx = F(b) - F(a)

where F(x) is the antiderivative of f(x).

Step-by-Step Evaluation of ∫₀⁵ (5x² + 25) dx

Let's now apply this knowledge to evaluate our specific definite integral: ∫₀⁵ (5x² + 25) dx.

Step 1: Find the Antiderivative

First, we need to find the antiderivative of the integrand, 5x² + 25. We do this by applying the power rule of integration, which states that the integral of xⁿ is (xⁿ⁺¹)/(n+1) + C, where C is the constant of integration.

Applying the power rule to each term:

• The antiderivative of 5x² is (5x³)/3.
• The antiderivative of 25 is 25x.

Therefore, the antiderivative of 5x² + 25 is (5x³)/3 + 25x + C. However, since we are dealing with a definite integral, the constant of integration (C) will cancel out during the subtraction step. So, we can omit it for now.

Step 2: Evaluate at the Upper Limit

Next, we substitute the upper limit of integration (5) into the antiderivative:

[(5(5)³)/3 + 25(5)] = [(5(125))/3 + 125] = (625/3) + 125 = (625 + 375)/3 = 1000/3

Step 3: Evaluate at the Lower Limit

Now, we substitute the lower limit of integration (0) into the antiderivative:

[(5(0)³)/3 + 25(0)] = 0

Step 4: Subtract the Lower Limit from the Upper Limit

Finally, we subtract the value of the antiderivative at the lower limit from the value at the upper limit:

(1000/3) - 0 = 1000/3

Therefore, the value of the definite integral ∫₀⁵ (5x² + 25) dx is 1000/3 or approximately 333.33.

Visualizing the Integral

It's helpful to visualize what this definite integral represents. The function f(x) = 5x² + 25 is a parabola that opens upwards. The definite integral calculates the area under this parabola between x = 0 and x = 5. This area is a combination of a rectangle (area = 25*5 = 125) and a segment under the parabola, giving a total area of 1000/3 square units.

Applications of Definite Integrals

Definite integrals have widespread applications across various fields, including:

• Physics: Calculating work done by a variable force, finding the center of mass of an object, determining the displacement of an object from its velocity function.
• Engineering: Calculating areas and volumes of irregular shapes, determining the stress and strain in materials, analyzing fluid flow.
• Economics: Calculating total revenue or cost over a period of time, finding consumer surplus or producer surplus.
• Probability and Statistics: Determining probabilities using probability density functions.

Advanced Integration Techniques (Beyond the Scope of this Specific Integral)

While the integral ∫₀⁵ (5x² + 25) dx was solved using the basic power rule, more complex integrals may require advanced techniques such as:

• Integration by substitution (u-substitution): Used to simplify integrals by substituting a new variable.
• Integration by parts: A technique for integrating products of functions.
• Partial fraction decomposition: Used to integrate rational functions.
• Trigonometric substitutions: Used to simplify integrals involving trigonometric functions.
• Numerical integration: Approximation techniques used when analytical solutions are difficult or impossible to obtain.

Conclusion

Evaluating definite integrals is a fundamental concept in calculus with wide-ranging applications. The step-by-step process, as demonstrated with the integral ∫₀⁵ (5x² + 25) dx, involves finding the antiderivative, evaluating it at the limits of integration, and then subtracting the lower limit value from the upper limit value. Understanding this process is crucial for anyone studying calculus or applying it in various fields. While this example used a relatively straightforward integral, the principles remain the same for more complex integrals, often requiring the application of more sophisticated integration techniques. Mastering definite integration is a key step towards a deeper understanding of calculus and its practical applications. This skill allows you to solve a vast array of problems across diverse scientific and engineering disciplines, solidifying its importance in mathematical education and research. Furthermore, the ability to visualize these integrals as areas under curves provides a powerful intuitive grasp of the underlying concepts, allowing for a more robust and flexible understanding of the subject matter.