91Ó°ÊÓ

One of the fundamental tools for solving integrals is the power rule for integration. This rule allows us to integrate functions of the form \({ x^n }\). The power rule states that if \({ n eq -1 }\), the integral of \({ x^n }\) with respect to \({ x }\) is given by: \[ \int x^n \, dx = \frac{x^{n+1}}{n+1} \] This formula is very useful because it transforms a polynomial expression into a simpler form we can work with. For example, in the original exercise, when we encounter \({ x^{-3} }\) and \({ x^{1/3} }\), we are able to apply the power rule to integrate each term separately. The new exponent will be \({ n + 1 }\) and we divide by this new exponent.

To make complex integrals easier, we can split them into smaller, more manageable parts. This is known as splitting integrals. In the original exercise, we first split the integral: \[ \int \left( \frac{7}{2 x^3} - \sqrt[3]{x} \right) \, dx \] into two separate integrals: \[ \int \frac{7}{2 x^3} \, dx - \int \sqrt[3]{x} \, dx \] This approach simplifies the problem and allows us to concentrate on each part individually. Once we have simpler integrals, we can apply the power rule or other methods to solve them.

When performing indefinite integrals, we always add a constant of integration, denoted by \({ C }\). This constant accounts for any constant terms that would disappear when differentiating. For example, if you have \({ F(x) }\) as an antiderivative of \({ f(x) }\), then \({ F(x) + C }\) is equally valid because: \[ \frac{d}{dx} (F(x) + C) = f(x). \] In the original exercise, after integrating both terms separately, we add the constant of integration at the end: \[ -\frac{7}{4} x^{-2} - \frac{3}{4} x^{4/3} + C \] This indicates the general family of functions that are solutions to the integrals we calculated. Including the constant of integration is crucial for accurately representing the complete set of solutions to the integral.