91Ó°ÊÓ

The definite integral is a fundamental concept in calculus. It represents the signed area under a function's curve within a specific interval. For the integral \( \int_{a}^{b} f(x) \, dx \), the values \( a \) and \( b \) are the bounds of integration. These bounds determine the specific range for which the area is calculated. The process essentially sums up infinitesimal slices across the interval.

• Lower and upper bounds: These are the integral's limits, symbolizing the region over which the function is integrated.
• Net area: Unlike indefinite integrals, definite integrals account for the direction, yielding positive and negative values based on the function's position relative to the x-axis.

In this exercise, the definite integral from \( 0 \) to \( \frac{\pi}{3} \) for \( \frac{\sin x}{(1+\cos x)^{2}} \) evaluates this signed area efficiently.

Trigonometric substitution is a technique used to simplify the integration of expressions involving square roots, fractions, or both. It's especially useful when dealing with functions containing terms like \( \sin x \) or \( \cos x \). By substituting trigonometric identities or parts of the integral, the expression becomes easier to integrate.
For instance, to solve \( \int_{0}^{\pi / 3} \frac{\sin x}{(1+\cos x)^{2}} \, dx \), we used the substitution \( u = 1 + \cos x \). This choice transforms the integral into a more manageable form by simplifying the function's denominator.

• Identify substitutions: Choose a substitution that simplifies the integral. Here, \( u = 1 + \cos x \) aligns terms with \( \sin x \), making the expression simpler.
• Change variables: Derivatives \( du = -\sin x \, dx \) help convert the integral into a new variable space, easing the integration process.

Through substitution, trigonometric functions that originally posed challenges can be re-expressed and integrated more straightforwardly.

Power rule integration is a foundational technique in calculus that aids in evaluating integrals involving powers of a variable. It states that for any real number \( n eq -1 \), the integral \( \int x^n \, dx \) is \( \frac{x^{n+1}}{n+1} + C \), with \( C \) being the constant of integration.
In the context of our exercise, once the integral \( \int \frac{1}{u^{2}} \, du \) is transformed into the form \( \int u^{-2} \, du \), the power rule can be directly applied.

• General form: Convert expressions, if necessary, to a straightforward power form \( x^n \).
• Application: The expression \( \int u^{-2} \, du \) becomes \( -u^{-1} = -\frac{1}{u} \) using the power rule.

Power rule integration turns complex polynomial expressions into manageable antiderivatives, simplifying the computation of definite integrals.

Change of variables, often referred to as substitution in integration, is crucial for transforming integrals into simpler forms. This technique involves introducing a new variable to replace an existing one, making the integral easier to solve.
In the example of \( \int_{0}^{\pi / 3} \frac{\sin x}{(1+\cos x)^{2}} \, dx \), changing variables by setting \( u = 1 + \cos x \), we effectively redirect the problem into a more solvable path.

• Establish new limits: When changing variables, it's crucial to adjust the integral's limits to match the new variable. Here, when \( x = 0 \), \( u = 2 \), and when \( x = \frac{\pi}{3} \), \( u = 1.5 \).
• Rewrite the integral: Substitute \( dx \) and adjust the differential appropriately. With \( du = -\sin x \, dx \), the transformation facilitates straightforward integration.

This approach is a versatile tool in calculus, simplifying seemingly difficult integrals into elementary antiderivatives.