91Ó°ÊÓ

In calculus, the definite integral represents the accumulation of quantities, where you calculate the area under a curve. It is expressed as:

• The integral sign \( \int \)
• A function \( f(x) \) called the integrand
• Differential \( dx \) indicating the variable of integration
• Two limits of integration, \( a \) and \( b \), showing the interval of integration

For our exercise, the definite integral is evaluated between the limits 5 and 6 for the function \( x(x-5)^{10} \). These boundaries help calculate the net area on the given interval.
The definite integral differs from the indefinite integral because it provides a numerical value rather than a function family. It's a critical part of integral calculus for solving real-world problems, from physics to engineering, by computing total amounts over a specific range.

Integration techniques are strategies to evaluate integrals, particularly when straightforward integration is difficult. Techniques like substitution, integration by parts, or partial fractions may be applied.
For our given integral \( \int_{5}^{6} x(x-5)^{10} dx \), substitution is a powerful method. Here, we use \( t = x - 5 \), which simplifies our integrand considerably. It transforms the integral to a form \( \int (t+5) t^{10} dt \), which is easier to handle.
This technique simplifies the integration process by changing the variable, making our integrals more approachable. Moreover, understanding various techniques is essential as they provide tools to tackle a wide range of complex integrals.

The power rule for integration is a fundamental tool in calculus. It simplifies the integration of functions with powers of the variable. For any function of the form \( x^n \), where \( n eq -1 \), the rule states: \[ \int x^n \, dx = \frac{x^{n+1}}{n+1} + C \]This rule makes it easy to integrate terms like \( t^{11} \) and \( t^{10} \) encountered in the exercise.
In our example, applying the power rule to \( t^{11} \) yields \( \frac{1}{12}t^{12} \), and to \( t^{10} \) gives \( \frac{5}{11}t^{11} \). It's crucial because it allows quick determination of antiderivatives, streamlining the entire integration process.
However, remember that it does not apply if the exponent is \(-1\), which leads to a logarithmic integration.

Antiderivatives, or indefinite integrals, are the reverse process of differentiation. Finding an antiderivative involves determining a function which derivative equals the given function.
In our exercise, finding the antiderivative allows solving the integral. When \( \int (t+5) t^{10} dt \) is calculated, we split it into two simpler integrals: \( \int t^{11} dt \) and \( \int 5t^{10} dt \).
The solution requires finding an antiderivative for each part using the power rule, resulting in the expression \( \frac{1}{12}t^{12} + t^{11} \). After this, computing the definite integral involves evaluating these expressions at the given limits and determining the net value.
The antiderivative is essential in determining the definite integral, embodying a fundamental principle of integral calculus that links integrals and derivatives.