91Ó°ÊÓ

The concept of a definite integral is foundational in calculus. A definite integral calculates the total accumulation of quantities, like area under a curve, between two bounds. When evaluating a definite integral, you're finding the net area between the curve of a function and the horizontal axis over a specified interval.
In the context of the given exercise, the definite integral \\( \int_{1}^{4} \frac{1}{\sqrt{x}(\sqrt{x}+1)^{3}} \, dx \)\ evaluates the function over the interval from 1 to 4. Here, our goal is to compute the total accumulated value of the function within this range.

• The limits of integration, 1 and 4, are essential in this calculation. They determine the start and end points for evaluation.
• The calculated result of the definite integral gives the net value over this specified interval.

Hence, the definite integral provides not only a way to integrate functions but also to apply this integration over specific ranges.

The substitution method is a technique used to simplify complex integrals by changing variables. It involves substituting part of the integrand with a new variable, which often turns the integral into a simpler form.
In our exercise, we used the substitution method where we let \\( u = \sqrt{x} \). This substitution leads to \\( x = u^2 \)\ and \\( dx = 2u \, du \). This change of variables simplifies the original integral into a more manageable form, allowing us to focus on solving a simpler expression.

• By substituting, we transformed the integral's limits of integration, changing from the original bounds for \\( x \)\ (1 to 4) to new bounds for \\( u \)\ (1 to 2).
• The substitution method is particularly useful for integrals that have composite functions or complex radicals.

This approach is powerful as it can turn a tough integral into something much easier to work with.

Integrating functions that are powers is a fundamental skill in calculus. The process involves using the power rule for integration, which states that the integral of \\( a^n \\ da \ = \frac{a^{n+1}}{n+1} + C \)\, provided that \\( n eq -1 \).
In the exercise, we rewrite the integral using a negative power, \\( (u+1)^{-3} \),which needs to be integrated. Let's look at how this power rule is applied:

• The expression \\( (u+1)^{-3} \)\ is incremented by one (i.e., -3 + 1 = -2).
• We divide by the new exponent, resulting in \\( \frac{(u+1)^{-2}}{-2} \),which simplifies and allows for further evaluation at the given bounds.

This power function integration plays a critical role in simplifying the integral to a form where specific limits are easily applied.

The bounds of integration are the limits within which a definite integral is evaluated. Changing variables can affect these bounds. Thus, tracking and adjusting them appropriately is vital.
For our integral, the original bounds of 1 to 4 on the variable \\( x \) become 1 to 2 on the new variable \\( u \) after substitution.

• The bounds indicate where to start and stop evaluating the integral, ensuring you're looking at the right section of the graph.
• After integration, these bounds help in applying the fundamental theorem of calculus by plugging in the upper and lower limits to find the net area.

Here, the immediate change due to substitution was updating these bounds, ensuring integration was accurate within the transformed variable context.This step is critical to prevent calculation errors as the variable and its interval transform together.