Evaluating Definite Integrals Step-by-Step A And B

In the realm of calculus, definite integrals play a crucial role in determining the accumulated value of a function over a specified interval. This article delves into the evaluation of two definite integrals, A and B, each involving rational functions. We will embark on a step-by-step journey, employing techniques such as polynomial long division, substitution, and the application of fundamental integration rules, to arrive at the precise values of these integrals. By the end of this exploration, you will gain a deeper understanding of the intricacies involved in evaluating definite integrals of this nature. Let's embark on this mathematical expedition together, unraveling the secrets held within these integrals.

We are tasked with evaluating the following definite integrals:

$$A = \int_{2}^{5} \frac{4x+0}{x+1} \, dx$$

$$B = \int_{2}^{5} \frac{4x}{x+1} \, dx$$

These integrals share a common integrand structure, featuring a rational function where the degree of the numerator is equal to the degree of the denominator. This characteristic necessitates the use of polynomial long division as a crucial initial step in the evaluation process. By performing this division, we can transform the integrand into a more manageable form, paving the way for the application of standard integration techniques. Let's delve into the solution process, unraveling the complexities of these integrals step by step.

Step 1: Polynomial Long Division

The integrand of integral A, $\frac{4x}{x+1}$, is an improper rational function because the degree of the numerator (1) is equal to the degree of the denominator (1). To simplify the integral, we perform polynomial long division:

 4
 x + 1 | 4x + 0
 - (4x + 4)
 --------
 -4

This division yields a quotient of 4 and a remainder of -4. Therefore, we can rewrite the integrand as:

$$\frac{4x}{x+1} = 4 - \frac{4}{x+1}$$

This transformation is a pivotal step in simplifying the integral, as it allows us to express the integrand as a sum of simpler terms that are readily integrable. The quotient, 4, is a constant term, and the remainder term, $\frac{-4}{x+1}$, is a rational function that can be easily integrated using a substitution. By performing this long division, we have effectively paved the way for a straightforward integration process.

Step 2: Rewriting the Integral

Now we can rewrite integral A using the result from the long division:

$$A = \int_{2}^{5} \left(4 - \frac{4}{x+1}\right) dx$$

This step is crucial as it transforms the original integral into a more manageable form, breaking it down into two simpler integrals. The first integral involves a constant term, while the second integral involves a rational function that can be easily integrated using a substitution. By separating the integral in this manner, we can apply standard integration techniques to each term individually, ultimately leading to the solution of the overall integral. This decomposition strategy is a common and effective approach when dealing with integrals of complex functions.

Step 3: Integrating Term by Term

We can now integrate term by term:

$$A = \int_{2}^{5} 4 \, dx - \int_{2}^{5} \frac{4}{x+1} \, dx$$

This step further simplifies the integration process by separating the integral into two distinct parts. The first part involves integrating a constant function, which is a straightforward application of the power rule of integration. The second part involves integrating a rational function, which can be tackled using a simple substitution. By breaking the integral down in this manner, we can apply the appropriate integration techniques to each part individually, making the overall evaluation process more manageable. This term-by-term integration strategy is a fundamental technique in calculus, allowing us to handle complex integrals by decomposing them into simpler components.

For the second integral, we use the substitution:

$$u = x + 1$$

$$du = dx$$

When $x = 2$, $u = 3$, and when $x = 5$, $u = 6$. The integral becomes:

$$\int_{2}^{5} \frac{4}{x+1} \, dx = 4 \int_{3}^{6} \frac{1}{u} \, du$$

This substitution is a key step in simplifying the integral of the rational function. By replacing the expression $x+1$ with a new variable $u$, we transform the integral into a simpler form that is readily integrable. The substitution also necessitates a change in the limits of integration, from the original $x$ values to the corresponding $u$ values. This transformation allows us to apply the standard integral of $\frac{1}{u}$, which is $\ln|u|$. The substitution technique is a powerful tool in calculus, enabling us to solve a wide range of integrals by simplifying the integrand and making it more amenable to integration.

Step 4: Evaluating the Integrals

Now we can evaluate the integrals:

$$A = \left[4x\right]_{2}^{5} - 4 \left[\ln|u|\right]_{3}^{6}$$

This step involves applying the fundamental theorem of calculus to evaluate the definite integrals. We first find the antiderivatives of the integrands and then evaluate them at the upper and lower limits of integration. The difference between these values gives us the definite integral. For the first term, the antiderivative of 4 is $4x$, and for the second term, the antiderivative of $\frac{1}{u}$ is $\ln|u|$. This step is a direct application of the fundamental theorem of calculus, a cornerstone of integral calculus that provides a powerful method for evaluating definite integrals. By carefully applying this theorem, we can arrive at the precise numerical value of the integral.

$$A = 4(5 - 2) - 4(\ln 6 - \ln 3)$$

$$A = 12 - 4 \ln \frac{6}{3}$$

$$A = 12 - 4 \ln 2$$

This step involves substituting the limits of integration into the antiderivatives and simplifying the expression. We first evaluate the antiderivatives at the upper limit of integration and then subtract the value of the antiderivatives at the lower limit of integration. This process yields a numerical value that represents the definite integral. In this case, we also use the property of logarithms that $\ln a - \ln b = \ln \frac{a}{b}$ to simplify the expression further. This step is a crucial part of the evaluation process, as it transforms the antiderivatives into a concrete numerical result, representing the accumulated value of the function over the specified interval.

Step 5: Final Result for A

Thus, the value of integral A is:

$$A = 12 - 4 \ln 2$$

This is the final result for integral A, representing the definite integral of the function $\frac{4x}{x+1}$ over the interval from 2 to 5. This value is a precise numerical representation of the accumulated area under the curve of the function within the given interval. The result is expressed in terms of a natural logarithm, which is a common occurrence when integrating rational functions. This final answer is the culmination of the step-by-step evaluation process, involving polynomial long division, substitution, and the application of the fundamental theorem of calculus.

Step 1: Polynomial Long Division

Integral B has the same integrand as integral A, so we already know from our previous calculation that:

$$\frac{4x}{x+1} = 4 - \frac{4}{x+1}$$

This reiterates the importance of polynomial long division when dealing with improper rational functions. By performing this division, we transform the integrand into a more manageable form, paving the way for integration. The result of the division, $4 - \frac{4}{x+1}$, is a crucial simplification that allows us to apply standard integration techniques. This step highlights the interconnectedness of mathematical concepts, as the result obtained in the previous problem directly applies to the current problem, saving us from redundant calculations.

Step 2: Rewriting the Integral

We can rewrite integral B as:

$$B = \int_{2}^{5} \left(4 - \frac{4}{x+1}\right) dx$$

This step mirrors the rewriting process in the solution for integral A, emphasizing the importance of expressing the integral in a form that is amenable to term-by-term integration. By separating the integrand into two simpler terms, we can apply the appropriate integration techniques to each term individually. This decomposition strategy is a powerful tool in integral calculus, allowing us to tackle complex integrals by breaking them down into smaller, more manageable components.

Step 3: Integrating Term by Term

Integrating term by term, we have:

$$B = \int_{2}^{5} 4 \, dx - \int_{2}^{5} \frac{4}{x+1} \, dx$$

This step, again, echoes the approach taken in solving integral A. It highlights the power of linearity in integration, allowing us to distribute the integral operator over sums and differences of functions. This property simplifies the integration process by enabling us to treat each term separately. The integral of a constant and the integral of the rational function can now be evaluated using standard techniques, making the overall integration process more efficient.

As before, we use the substitution:

$$u = x + 1$$

$$du = dx$$

When $x = 2$, $u = 3$, and when $x = 5$, $u = 6$. The integral becomes:

$$\int_{2}^{5} \frac{4}{x+1} \, dx = 4 \int_{3}^{6} \frac{1}{u} \, du$$

This substitution is identical to the one used in solving integral A, further emphasizing the consistency in the solution process. The substitution simplifies the integral of the rational function, transforming it into a standard integral that is readily evaluated. The change in limits of integration, from $x$ values to $u$ values, is crucial for accurate evaluation of the definite integral. This step reinforces the importance of recognizing patterns and applying previously learned techniques to new problems.

Step 4: Evaluating the Integrals

Now we evaluate the integrals:

$$B = \left[4x\right]_{2}^{5} - 4 \left[\ln|u|\right]_{3}^{6}$$

This step directly applies the fundamental theorem of calculus, just as in the solution for integral A. We evaluate the antiderivatives at the limits of integration and take the difference to obtain the definite integral. This step is a key application of the fundamental theorem, demonstrating its power in evaluating definite integrals of various functions. The careful and accurate application of this theorem is essential for arriving at the correct numerical result.

$$B = 4(5 - 2) - 4(\ln 6 - \ln 3)$$

$$B = 12 - 4 \ln \frac{6}{3}$$

$$B = 12 - 4 \ln 2$$

This simplification process mirrors the steps taken in solving integral A, highlighting the consistency in the mathematical operations. The substitution of the limits of integration and the application of logarithmic properties lead to a concise expression for the value of the integral. This step demonstrates the importance of algebraic manipulation in simplifying mathematical expressions and arriving at the final answer.

Step 5: Final Result for B

Thus, the value of integral B is:

$$B = 12 - 4 \ln 2$$

This is the final result for integral B, which is identical to the result for integral A. This outcome is not surprising, as the integrands of the two integrals are essentially the same. This result reinforces the importance of careful observation and recognizing similarities between problems, as it can lead to significant efficiencies in the solution process.

In conclusion, both definite integrals A and B evaluate to the same value, 12 - 4ln 2. This outcome underscores the importance of recognizing underlying similarities in mathematical problems, even when they appear different at first glance. The solutions involved the application of polynomial long division, substitution, and the fundamental theorem of calculus, demonstrating the interconnectedness of these techniques in solving integral problems. By mastering these techniques, one can effectively tackle a wide range of definite integral evaluations. This exploration serves as a testament to the power and elegance of calculus in solving real-world problems and furthering our understanding of the mathematical universe.