91Ó°ÊÓ

In calculus, the power rule for integration is a straightforward and essential technique. It allows us to calculate the indefinite integral of terms within the form of a power function, namely \( ax^n \). This rule is a counterpart to the power rule for differentiation that is foundational for a lot of calculus.

The formula used in integration states: the integral of \( ax^n \) is\[ \\int ax^n \, dx = \frac{a}{n+1}x^{n+1} + C \]\where:

• \( a \) is any constant,
• \( n \) is any real number except for -1,
• \( C \) is the constant of integration, more about this in the next section.

In the specific problem, this rule helps us determine the antiderivative of \( 5x^{-3} \) effectively. By identifying:

• \( a = 5 \)
• \( n = -3 \)

We apply the method to find the integral as:\[ -\frac{5}{2}x^{-2} + C \] \simplifying to \( -\frac{5}{2x^2} + C \).
Essentially, this rule gives us a formula for success when dealing with power functions in integration.

The constant of integration is a crucial concept in calculus. When finding an indefinite integral, it represents a constant added to the integral that accounts for the family of all possible antiderivatives.

Whenever we integrate a function, there is an unlimited number of antiderivatives we can find because a constant disappears during differentiation. Therefore, the constant of integration, denoted as \( C \), is included in the solution to represent these multiple possibilities.
For instance, if the derivative of \( f(x) = -\frac{5}{2x^2} \) was needed, any added constant would not appear in the derivative, as the derivative of a constant is zero.
This constant is what makes indefinite integrals different from definite integrals, which have specific limits and thus a specific antiderivative value.

After finding the indefinite integral, it's always a good idea to verify the results using differentiation. This step ensures you have not made any calculation mistakes, and it helps solidify your understanding.

In our example, once the integral \( -\frac{5}{2x^2} + C \) was found, we then differentiate it.
Differentiation involves applying rules to find a function's rate of change, effectively reversing integration.

The process involves the following:

• The derivative of \( -\frac{5}{2x^2} \) is calculated using the power rule for differentiation, confirming it returns to \( 5x^{-3} \), the original function.
• The constant \( C \)'s derivative is zero, which confirms its correct inclusion in integration.

Our correct retraction to the original function \( 5x^{-3} \) shows the integral's accuracy. Differentiation verification thus confirms both the method and the arithmetic used in integration are accurate, showcasing calculus's elegance in balancing operations.