91Ó°ÊÓ

The **substitution method** is a vital technique in calculus that helps simplify integrals. This method involves changing variables to turn a complex integral into a simpler one that is easier to solve. In our problem, we have the integral \( \int \sqrt[3]{2x-4} \, dx \).
To use substitution, we first choose a new variable to replace a part of the integrand. Here, let \( u = 2x - 4 \). This substitution captures the expression inside the cube root. Next, we differentiate \( u \) to find \( dx \). The differentiation gives \( du = 2 \, dx \), leading to \( dx = \frac{du}{2} \). This relationship allows us to replace \( dx \) in the integral.
With \( u \) and \( dx \) expressed in terms of \( du \), the original integral simplifies significantly to \( \int \sqrt[3]{u} \cdot \frac{du}{2} \). Substitution transforms the problem, turning the integral into a form that is more manageable for further calculations. This process is often the first step in solving complex integrals effectively.

**Integrand simplification** is a crucial step when solving integrals, especially after using substitution. Once we substitute the variables, the goal is to simplify the new integral. The complexity of the original integrand, \( \sqrt[3]{2x-4} \), is reduced after substitution to \( \sqrt[3]{u} \). During this process, constants outside the integral can be factored out to make further integration steps easier.
In our example, the integral \( \int \sqrt[3]{u} \cdot \frac{du}{2} \) can be rewritten as \( \frac{1}{2} \int u^{1/3} \, du \). Factoring out \( \frac{1}{2} \) is important because it simplifies the expression, allowing us to focus solely on integrating \( u^{1/3} \). By successfully simplifying the integrand, we make subsequent integration steps more straightforward and less prone to errors.

**Power rule integration** is one of the fundamental techniques for solving integrals, especially after simplification. It applies to functions of the form \( u^n \), making it highly useful in our example.
Using the power rule, \( \int u^n \, du = \frac{u^{n+1}}{n+1} + C \), we can integrate \( u^{1/3} \). In this scenario, \( n = \frac{1}{3} \). Applying the power rule results in \( \frac{u^{1/3+1}}{1/3+1} + C \). This simplifies to \( \frac{u^{4/3}}{4/3} \) after combining like terms.
Finally, we multiply by the factor \( \frac{1}{2} \) that we factored out previously, yielding \( \frac{1}{2} \cdot \frac{3u^{4/3}}{4} \), which reduces to \( \frac{3u^{4/3}}{8} + C \). This process highlights how applying the power rule maintains the methodical approach needed to solve integrals effectively. Once completed, the last step is to back-substitute the original expression for \( u \), resulting in the final solution expressed in terms of \( x \).