Evaluating the Integral ∫ from 1 to 4 of e^sqrt{x}/sqrt{x} dx using Substitution

In this article, we focus on the integral problem given and demonstrate a detailed step-by-step solution using the substitution method. We introduce the concept, provide a clear explanation, and walk the reader through the solution process. This article is suitable for students and professionals in mathematics and related fields who are looking to enhance their understanding of integral calculus.

Introduction

In the context of integral calculus, substitution (or u-substitution) is a powerful technique to simplify complex integrands. This method aims to transform the integral into a more manageable form. In this article, we tackle the problem of evaluating the integral (int_{1}^{4}{frac{e^{sqrt{x}}}{sqrt{x}}dx}). We'll illustrate the process step-by-step using the substitution (u sqrt{x}).

Step 1: Substitution and Differential Conversion

We start by making the substitution (u sqrt{x}). This substitution allows us to simplify the integrand. To maintain the integrity of the integral, we must also convert the differential (dx) in terms of (u).

Given:

(u sqrt{x}) implies u sqrt{x} iff du frac{dx}{2sqrt{x}})

Step 2: Expressing dx in Terms of u

We manipulate the substitution to express (dx) in terms of (u): du frac{dx}{2sqrt{x}} implies dx 2sqrt{x}du)

Step 3: Revising the Integral

Now, we substitute the expressions for (sqrt{x}) and (dx) in the integral. Recalling that (sqrt{x} u) and substituting (dx 2u du), we obtain: int_{1}^{4}{frac{e^{sqrt{x}}}{sqrt{x}}dx} int_{1}^{4} frac{e^{sqrt{x}}}{sqrt{x}} 2sqrt{x} du 2int_{1}^{4} e^{sqrt{x}} du))

Simplifying the expression, we get: 2int_{1}^{4} e^{sqrt{x}} du 2int_{1}^{2} e^u du))

Step 4: Evaluating the Integral

The integral on the right-hand side can be easily evaluated:

2int_{1}^{2} e^u du 2e^u Bigg|_{1}^{2} 2(e^2 - e)

Therefore, the solution to the integral is:

2(e^2 - e))

Converting this to a numerical value, we get approximately:

2(e^2 - e) approx 2(7.389 - 2.718) approx 9.342)

Conclusion

Through the substitution method and careful manipulation of differentials, we were able to evaluate the integral (int_{1}^{4}{frac{e^{sqrt{x}}}{sqrt{x}}dx}) to (2(e^2 - e)), approximately 9.342. This method showcases the power of substitution in simplifying complex integrands and provides a clear, step-by-step approach for understanding and solving integral problems.