Understanding the Evaluation of Integrals: A Step-by-Step Guide with Examples

Integrals are fundamental to calculus, providing a way to determine the area under a curve or to solve complex mathematical problems. One common method of evaluating integrals involves the use of substitution. This article will explore a specific integral problem and explain the step-by-step process of evaluating it. By the end, you will have a clear understanding of how to apply this technique in similar problems.

Problem Setup

Consider the following integral:

(int frac{sqrt{ln x}}{x} dx)

This integral may seem intimidating at first, but it can be simplified using a substitution method. Let's break down the process.

Step-by-Step Solution

Step 1: Substitution

The first step is to make a substitution that simplifies the integral. Let's start by setting

(u ln x)

Using this substitution, we can find the differential (du):

(du frac{1}{x} dx)

Now, we can replace (frac{1}{x} dx) with (du) in the integral:

(int sqrt{u} du)

Step 2: Integration

The integral of (sqrt{u}) is a standard integral. Recall that (sqrt{u} u^{frac{1}{2}}). Using the power rule for integration, we have:

(int u^{frac{1}{2}} du frac{u^{frac{3}{2}}}{frac{3}{2}} C frac{2}{3} u^{frac{3}{2}} C)

Substituting back (u ln x), we obtain the final answer:

(frac{2}{3} (ln x)^{frac{3}{2}} C)

Conclusion

By using the substitution method, we transformed the original integral into a simpler form that could be easily integrated. This process is a powerful technique that can be applied to many other integrals. Remember to always choose a substitution that simplifies the integral, and practice is key to mastering this technique.

Additional Resources

For further practice and to deepen your understanding, consider exploring the following resources:

Online integral calculators that can help you verify your solutions and provide step-by-step explanations. Math courses and tutorials on platforms like Khan Academy and Coursera, which cover integral calculus in detail. Practice problems in textbooks dedicated to calculus and integral calculus.

Understanding the evaluation of integrals is crucial for advanced mathematical studies, and this guide should provide you with the knowledge and practice needed to tackle more complex problems confidently.