Understanding U-Substitution in Integral Calculus

The method of u-substitution simplifies complex integrals into manageable forms.

### Understanding U-Substitution in Indefinite Integrals
* 📚 The integral simplifies to \( U^3 \, du \), where the anti-derivative of \( U^3 \) is **\( \frac{U^4}{4} + C \)**.
* 🔄 By replacing \( U \) with \( x + 1 \), we derive the **final answer** from the **anti-derivative**.
* 💡 This lecture aims to clarify the method of **u-substitution**, a valuable technique for **solving integrals**.
* 🤔 The substitution method involves choosing a function \( G(x) \) and letting \( u = G(x) \), where \( f \) must be **continuous on the range of \( G \)**.
* 🔍 The differential \( du \) becomes **\( G'(x) \, dx \)** when rearranged, allowing us to substitute **\( G(x) \)** and **\( G'(x) \, dx \)** in the integral.
* 📝 For example, if we set \( u = x + 1 \), then \( du = dx \), leading to a **simplified integral**.
* ✨ Overall, this technique **streamlines the process of finding indefinite integrals**, making it **easier to handle complex functions**.
* 👍 If you found this lecture helpful, consider **liking and subscribing** for more insightful content on integral techniques.

In this lecture, we'll introduce the method of substitution and find a simple indefinite integral using the substitution method. The first question is, what is your substitution method? Let's consider an indefinite integral of the following form, where f is a continuous function on the range of G of X. Recall that we need this condition on the function f at the minimum so that we have an integral function. Then, by simply letting u to be x, d over DX becomes G Prime of X. Now, by moving DX to the other side of the equation, we get the DU is G Prime of X DX. Then, by simply replacing G of X with u in the original integral and G Prime of X DX with the U, we finally arrive at the following form and this beautiful theorem. So, how can this theorem be applied? Let's consider the following very simple example here. Let's let x + 1 to be U. Then, its derivative du over DX becomes just one, which follows that du is DX. By replacing x + one in the original integral with u and DX with du, the original integral

simplifies to the integral of U to the 3rd power du. Now using the power rule for integrals, the anti-derivative of U to the 3rd power is U to the 4th power divided by 4 plus a constant C. All you have to do now is replace U with x + one, and we arrive at this final answer. It wasn't too bad, was it? I hope this video helped you understand one of the most useful integral techniques, namely the method of u-substitution. If you enjoyed this lecture, be sure to like and subscribe to my channel for more useful videos.

Calculus 2 : The method of u substitution (Theory)

Learn with Dr. Charles Kang

Introduction to the Substitution Method for Indefinite Integrals

The substitution method is used to find indefinite integrals by letting u = G(X) and relating du to DX.

This method requires the function f to be continuous on the range of G(X) for the integral to exist.

By replacing G(X) with u and DX with du, the integral can be simplified, leading to a more manageable form for calculation.

The integral of U to the 3rd power simplifies to U to the 4th power divided by 4, plus a constant C.

Replacing U with x + one gives the final answer.

The video aims to teach the method of u-substitution for integrals and encourages viewers to like and subscribe for more content.