91Ó°ÊÓ

In calculus, trigonometric identities are helpful tools that simplify complex expressions by rewriting them in equivalent forms. They are used extensively in trigonometric integration problems to make integrals easier to solve.
One of the identities used often is the Pythagorean identity, which states: \\( \sin^2 x + \cos^2 x = 1 \).
This identity allows us to replace one trigonometric function with another. In this exercise, the identity is used to rewrite the denominator, originally presented as \( 1 + \sin^2 x \). By using the identity, the expression \( \cos^2 x + \sin^2 x \) simplifies to 1, showcasing the advantage of trigonometric identities in simplifying the terms in the integral.
Understanding these identities can help in recognizing opportunities to simplify terms and make integration more approachable.

Integration techniques are methods that are employed to solve integrals, particularly when they are not straightforward.
Some common techniques include substitution, integration by parts, and trigonometric substitution. In this specific problem, we utilize the simplification of the integrand as an integration technique.
By recognizing the trigonometric identity \( \cos^2 x + \sin^2 x = 1 \), we simplify the integral, turning it from \(\frac{\cos x}{\cos^2 x + \sin^2 x}\) into just \( \cos x \).
This simplification makes the integration much more direct. Once simplified, integrating \( \cos x \) is straightforward because the antiderivative of \( \cos x \) is known to be \( \sin x \). Thus, recognizing when and how to apply specific techniques is crucial in evaluating integrals efficiently.

The limits of integration, or bounds, indicate the range over which the integration is performed. For definite integrals, these bounds transform an indefinite integral into a computation of the accumulation of area under a curve between two points.
In our exercise, the integral runs from 0 to \( \pi/2 \), covering one quadrant of the unit circle.
Evaluating the definite integral involves substituting these limits into the antiderivative. Once the integral simplifies to \( \sin x \), the next step is to evaluate it at these limits.
By plugging in the upper limit \( \pi/2 \), we get \( \sin(\pi/2)\), and similarly for the lower limit 0, we get \( \sin(0) \).

• The final value of the integral: \( \sin(\pi/2) - \sin(0) \), imposes understanding of how limits affect the solution.

Understanding limits of integration helps solidify how definite integrals calculate the net area under a curve.