Solving The Integral: ∫(2x+1)/(x^2+x-1) Dx From 1 To 3

Let's dive into the world of calculus and tackle the problem of evaluating the definite integral ∫(2x+1)/(x^2+x-1) dx from 1 to 3. This type of problem often appears in calculus courses and requires a solid understanding of integration techniques, particularly u-substitution. In this comprehensive guide, we will break down the steps to solve this integral, ensuring you grasp the underlying concepts. Understanding the problem clearly is the first step towards finding the solution. When dealing with integrals of rational functions, it’s often useful to check if the numerator is related to the derivative of the denominator, which can significantly simplify the integration process. We aim to provide a clear, step-by-step explanation, making the process accessible even if you're just beginning your calculus journey.

1. Recognizing the Potential for u-Substitution

When we encounter an integral like ∫(2x+1)/(x^2+x-1) dx, the first thing to consider is whether u-substitution can be applied. U-substitution is a powerful technique that simplifies integrals by replacing a complex expression with a single variable, making the integral easier to solve. The key to identifying whether u-substitution is suitable lies in recognizing a function and its derivative within the integral. In our case, observe that the derivative of the denominator, x^2 + x - 1, is 2x + 1, which appears in the numerator. This is a strong indicator that u-substitution will be an effective method.

To formally apply u-substitution, we let u = x^2 + x - 1. This choice is crucial because it simplifies the denominator, turning a quadratic expression into a single variable. Next, we find the derivative of u with respect to x, denoted as du/dx. The derivative of x^2 is 2x, the derivative of x is 1, and the derivative of the constant -1 is 0. Therefore, du/dx = 2x + 1. Now, we express du in terms of dx, which gives us du = (2x + 1) dx. This expression is exactly what we have in the numerator of our integral, making the substitution straightforward. Recognizing these relationships is a fundamental skill in calculus, allowing you to transform complex integrals into simpler forms. By choosing the correct substitution, we set the stage for an easier integration process.

2. Performing the u-Substitution

Now that we've identified u = x^2 + x - 1 and du = (2x + 1) dx, we can substitute these into the original integral. This process transforms the integral from being in terms of x to being in terms of u, which simplifies the expression significantly. Substituting u and du into the integral ∫(2x+1)/(x^2+x-1) dx, we replace the denominator x^2 + x - 1 with u and the numerator (2x + 1) dx with du. This gives us a new integral: ∫(1/u) du. This transformed integral is much simpler to handle. It is a basic integral that we readily recognize.

The integral of 1/u with respect to u is the natural logarithm of the absolute value of u, denoted as ln|u|. The absolute value is crucial because the natural logarithm is only defined for positive values. Therefore, ∫(1/u) du = ln|u| + C, where C is the constant of integration. Remember, when dealing with indefinite integrals, we always include the constant of integration to account for the family of functions that have the same derivative. The u-substitution has effectively turned a complex rational function integral into a basic logarithmic integral. This step highlights the power of substitution in simplifying calculus problems, making them more manageable and easier to solve. By carefully choosing the substitution, we've reduced the problem to a fundamental integration, showcasing the elegance of calculus techniques.

3. Changing the Limits of Integration

Since we are evaluating a definite integral, we need to consider the limits of integration. The original limits were in terms of x (from x = 1 to x = 3), but now our integral is in terms of u. Therefore, we must change the limits of integration to correspond to the new variable u. To do this, we use the substitution equation, u = x^2 + x - 1. When x = 1, we substitute this value into the equation to find the corresponding value of u: u = (1)^2 + (1) - 1 = 1. So, the lower limit of integration in terms of u is 1. Next, we find the upper limit of integration by substituting x = 3 into the equation: u = (3)^2 + (3) - 1 = 9 + 3 - 1 = 11. Therefore, the upper limit of integration in terms of u is 11.

Changing the limits of integration is a critical step in evaluating definite integrals using substitution. If we don't change the limits, we would need to substitute back for x after integrating, which can sometimes be more complicated. By converting the limits to the new variable, we can directly evaluate the integral in terms of u, making the process more streamlined. Now, our integral is ∫(1/u) du with limits from 1 to 11. This transformation ensures that our final answer is accurate and consistent with the original problem. This step underscores the importance of adapting all components of the integral, including the limits, when performing substitutions in definite integrals. With the correct limits, we are ready to evaluate the integral in its simplified form.

4. Evaluating the Integral

With the integral now in the form ∫(1/u) du and the limits of integration changed to 1 and 11, we are ready to evaluate the integral. As we established earlier, the integral of 1/u with respect to u is ln|u|. Therefore, we have ∫(1/u) du = ln|u|. Now, we need to evaluate this at the limits of integration, which are 1 and 11. To evaluate a definite integral, we find the antiderivative (in this case, ln|u|) and then subtract the value of the antiderivative at the lower limit from the value at the upper limit. This is the Fundamental Theorem of Calculus in action.

First, we evaluate ln|u| at the upper limit, u = 11. This gives us ln|11|, which is simply ln(11) since 11 is positive. Next, we evaluate ln|u| at the lower limit, u = 1. This gives us ln|1|, which is ln(1). The natural logarithm of 1 is 0, so ln(1) = 0. Now, we subtract the value at the lower limit from the value at the upper limit: ln(11) - ln(1) = ln(11) - 0 = ln(11). Thus, the value of the definite integral ∫(1/u) du from 1 to 11 is ln(11). This step demonstrates the final application of the Fundamental Theorem of Calculus, providing the exact value of the definite integral. The simplicity of the result, ln(11), highlights the effectiveness of u-substitution in transforming a seemingly complex integral into a manageable form.

5. The Final Answer

After performing the u-substitution, changing the limits of integration, and evaluating the integral, we arrive at the final answer. The definite integral ∫(2x+1)/(x^2+x-1) dx from 1 to 3 is equal to ln(11). This result represents the exact value of the integral, providing a precise solution to the problem. The journey from the initial complex integral to the final simple answer showcases the power of calculus techniques in simplifying and solving mathematical problems.

To summarize, we started by recognizing the potential for u-substitution, identifying that the numerator was related to the derivative of the denominator. We then performed the substitution, transforming the integral into a simpler form. Next, we changed the limits of integration to reflect the new variable, ensuring accurate evaluation. We evaluated the integral using the Fundamental Theorem of Calculus, and finally, we arrived at the solution: ln(11). This step-by-step process not only solves the problem but also reinforces the key concepts and techniques of integral calculus. The final answer, ln(11), is a testament to the effectiveness of these methods in finding precise solutions to definite integrals.

In conclusion, evaluating the definite integral ∫(2x+1)/(x^2+x-1) dx from 1 to 3 involves a series of strategic steps, primarily utilizing u-substitution. The result, ln(11), is a concise and elegant solution that underscores the power and beauty of calculus. For further exploration of calculus concepts and techniques, you might find valuable resources at Khan Academy Calculus. 😜