91Ó°ÊÓ

The power rule of integration is a fundamental tool in calculus, particularly for integrating polynomials. This rule helps simplify the process of finding indefinite integrals. It states that to integrate a term of the form \( x^n \), where \( n eq -1 \), you increase the exponent by 1, then divide by the new exponent.
For example, the integration of \( x^n \) is given by:

• \( \int x^n \, dx = \frac{x^{n+1}}{n+1} + C \)

Here, \( C \) represents the constant of integration, which is added because integrals can have infinite possible solutions differing by a constant.
This method makes it easier to handle polynomial expressions where each term can be integrated separately using the power rule.

Various integration techniques are available to tackle different functional forms. In this exercise, simplifying the integrand is a crucial first step. Simplification often involves algebraic manipulation, such as dividing terms or reducing radicals.
For example, dividing each term of the polynomial by the denominator \( x \) in the integrand \( \frac{x^{3}-2 \sqrt{x}}{x} \) simplifies it to \( x^2 - 2x^{-1/2} \).
This simplification transforms the problem into a form suitable for direct application of integration rules. By addressing each term independently, we can focus on applying the appropriate technique, like the power rule, to find the indefinite integral.
These techniques can be used as building blocks to tackle more complex integrals.

Polynomial integration deals with integrating expressions where each term is a power of \( x \). These are generally straightforward due to the direct applicability of the power rule.
In the given exercise, after simplifying the expression \( \frac{x^3}{x} - \frac{2\sqrt{x}}{x} \) to \( x^2 - 2x^{-1/2} \), we can individually integrate each term:

• For \( x^2 \), integration results in \( \frac{x^3}{3} \).
• For \(-2x^{-1/2} \), integration involves adjusting the exponent and results in \(-4x^{1/2} \).

Finally, the integrated terms are combined to form the complete indefinite integral. Each polynomial term's integration adds up to give the final expression \( \frac{x^{3}}{3} - 4x^{1/2} + C \).
This demonstrates the effectiveness and simplicity of polynomial integration for simplifying and solving indefinite integrals.