Calculate The Definite Integral Of (2 - 5/√(5x + 4)) Dx From 0 To 1.

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Introduction

In this article, we will delve into the process of calculating the definite integral of the function (2 - 5/√(5x + 4)) over the interval from 0 to 1. This problem falls under the category of calculus, specifically integral calculus, and requires us to find the antiderivative of the given function and then evaluate it at the limits of integration. The definite integral represents the area under the curve of the function between the specified limits. Understanding how to compute definite integrals is crucial in various fields such as physics, engineering, and economics, where it is used to calculate quantities like displacement, work, and accumulated change.

Understanding the Problem

Before we dive into the solution, let's break down the problem. We are asked to evaluate the definite integral:

∫[0 to 1] (2 - 5/√(5x + 4)) dx

This means we need to find a function F(x) such that its derivative F'(x) is equal to (2 - 5/√(5x + 4)). Once we find F(x), we will evaluate it at the upper limit (1) and the lower limit (0) and subtract the latter from the former. This process is based on the Fundamental Theorem of Calculus, which establishes the relationship between differentiation and integration. The integrand, (2 - 5/√(5x + 4)), consists of two terms: a constant term (2) and a more complex term (-5/√(5x + 4)). We will need to handle each term separately and then combine the results.

Step-by-Step Solution

1. Split the Integral

The first step is to split the integral into two separate integrals, one for each term in the integrand. This is allowed due to the linearity property of integrals, which states that the integral of a sum (or difference) is the sum (or difference) of the integrals. So, we have:

∫[0 to 1] (2 - 5/√(5x + 4)) dx = ∫[0 to 1] 2 dx - ∫[0 to 1] 5/√(5x + 4) dx

This simplifies the problem by allowing us to focus on each term individually. The first integral, ∫[0 to 1] 2 dx, is straightforward and can be easily evaluated. The second integral, ∫[0 to 1] 5/√(5x + 4) dx, requires a bit more work, specifically a u-substitution.

2. Evaluate the First Integral

The integral of a constant function is simply the constant times the variable of integration. In this case, the integral of 2 with respect to x is 2x. So, we have:

∫[0 to 1] 2 dx = [2x][0 to 1]

Now, we evaluate 2x at the upper and lower limits:

[2(1) - 2(0)] = 2 - 0 = 2

Thus, the first part of the definite integral is 2. This represents the area of a rectangle with a height of 2 and a width of 1, which is intuitively 2 square units.

3. Evaluate the Second Integral Using u-Substitution

The second integral, ∫[0 to 1] 5/√(5x + 4) dx, requires a u-substitution to simplify the integrand. Let's set:

u = 5x + 4

Then, we find the derivative of u with respect to x:

du/dx = 5

Solving for dx, we get:

dx = du/5

Now, we substitute u and dx into the integral:

∫ 5/√(5x + 4) dx = ∫ 5/√u (du/5) = ∫ 1/√u du = ∫ u^(-1/2) du

We also need to change the limits of integration to be in terms of u. When x = 0, u = 5(0) + 4 = 4. When x = 1, u = 5(1) + 4 = 9. So, the new limits of integration are 4 and 9.

Now, we can rewrite the integral with the u-substitution and the new limits:

∫[0 to 1] 5/√(5x + 4) dx becomes ∫[4 to 9] u^(-1/2) du

4. Find the Antiderivative

To find the antiderivative of u^(-1/2), we use the power rule for integration, which states that ∫x^n dx = (x^(n+1))/(n+1) + C, where C is the constant of integration. Applying this rule, we get:

∫ u^(-1/2) du = (u^(-1/2 + 1))/(-1/2 + 1) + C = (u^(1/2))/(1/2) + C = 2√u + C

5. Evaluate the Antiderivative at the Limits of Integration

Now, we evaluate the antiderivative at the limits of integration (4 and 9):

[2√u][4 to 9] = 2√9 - 2√4 = 2(3) - 2(2) = 6 - 4 = 2

So, the value of the second integral is 2.

6. Combine the Results

Finally, we combine the results from the two integrals:

∫[0 to 1] (2 - 5/√(5x + 4)) dx = ∫[0 to 1] 2 dx - ∫[0 to 1] 5/√(5x + 4) dx = 2 - 2 = 0

Therefore, the definite integral of (2 - 5/√(5x + 4)) from 0 to 1 is 0.

Verification and Conclusion

We have systematically computed the definite integral of the given function over the specified interval. The process involved splitting the integral, using u-substitution, finding antiderivatives, and evaluating at the limits of integration. The final result of 0 indicates that the net area under the curve of the function (2 - 5/√(5x + 4)) between x = 0 and x = 1 is zero. This could mean that the areas above and below the x-axis cancel each other out within this interval. The result can be verified using computational tools or software that can perform symbolic integration. This problem illustrates the application of the Fundamental Theorem of Calculus and the importance of techniques like u-substitution in solving integrals.

In conclusion, by carefully applying the rules and techniques of integral calculus, we have successfully calculated the definite integral of (2 - 5/√(5x + 4)) from 0 to 1 and found it to be 0. This exercise not only reinforces the understanding of integral calculus but also highlights the importance of these concepts in practical applications.