91Ó°ÊÓ

When evaluating an integral, such as \( \int_{1}^{2} \frac{1}{x^{6}} \, dx \), we are essentially seeking the total accumulation of the function \( \frac{1}{x^{6}} \) over the interval from \( x = 1 \) to \( x = 2 \). This process involves finding the net area under the curve of the function within this range.
To perform an integral evaluation, follow these crucial steps:

• Identify the bounds of integration, which in this case are \( 1 \) and \( 2 \).
• Ensure the function inside the integral, here \( \frac{1}{x^{6}} \), is correct and manageable.
• Find the antiderivative using a suitable integration method, like the power rule for this problem.
• Apply the Fundamental Theorem of Calculus to evaluate the definite integral.

By understanding these steps, integral evaluations can be tackled more effectively, unlocking solutions to problems that model real-world scenarios involving rates of change and accumulations.

An antiderivative of a function is another function whose derivative equals the original function. It can be considered a reverse derivative. For instance, in our problem, looking for an antiderivative of \( f(x) = \frac{1}{x^{6}} \), you would find a function \( F(x) \) such that \( F'(x) = f(x) \).
Finding an antiderivative involves integration, and to do this correctly, let’s consider:

• The function and its forms: Rewrite \( f(x) = \frac{1}{x^{6}} \) as \( x^{-6} \) for easier manipulation.
• Using integration rules: Apply the power rule in integration to find the antiderivative.

For \( x^{-6} \), the antiderivative is found using the relation \( \int x^n \, dx = \frac{x^{n+1}}{n+1} + C \). Here, \( n = -6 \), leading to \( \int x^{-6} \, dx = \frac{x^{-5}}{-5} + C \), which simplifies to \( F(x) = -\frac{1}{5x^{5}} \). This analytical form represents the accumulated area from a starting point to any point \( x \), ultimately vital for definite integration.

The power rule for integration is a key technique in calculus that simplifies finding the antiderivative of a function. It's expressed as \( \int x^n \, dx = \frac{x^{n+1}}{n+1} + C \), where \( n eq -1 \). This rule transforms a power of \( x \) by increasing its exponent by 1, and then dividing by the new exponent.
In our illustrative problem, we use the power rule for \( x^{-6} \):

• Increment the exponent: From \( -6 \) to \( -5 \).
• Divide the transformed function by the new exponent: \( \frac{x^{-5}}{-5} \).

This results in the antiderivative \( F(x) = -\frac{1}{5x^{5}} \), a crucial step in solving the integral \( \int_{1}^{2} \frac{1}{x^{6}} \, dx \). Utilizing the power rule simplifies potentially complex integrations, especially when dealing with polynomials or powers of \( x \). It's one of the foundational tools in calculus, facilitating the transition from the derivative of functions to their antiderivatives efficiently.