91Ó°ÊÓ

Trigonometric substitution is a powerful technique used to evaluate integrals, especially when dealing with radicals involving squares or expressions containing trigonometric functions. This approach involves substituting a trigonometric function for a variable, simplifying the integral into a form that is easier to evaluate.

In this exercise, we used trigonometric substitution in two forms: first, by letting \( u = \cos x \) and then by \( u = \sin x \). These substitutions transformed the original integral \( \int \sin x \cos x \, dx \) into manageable expressions of \( \int -u \, du \) and \( \int u \, du \), respectively.

With simplification, both approaches funnel into the result \( \frac{1}{2}\sin^2 x + C \). This demonstrates how trigonometric substitution can sometimes offer multiple pathways to a solution that eventually converges to the same result.

Integration by parts is a valuable tool for integrals that are products of functions. This technique is based on the formula:

• \( \int u \, dv = uv - \int v \, du \)

For the integral \( \int \sin x \cos x \, dx \), we assign \( u = \sin x \) and \( dv = \cos x \, dx \). This choice leads us through a transformation process where we use the formula to ultimately break down the product into simpler components.

The initial integration by parts gives an equation involving the original integral, yielding a solvable equation: \( 2\int \sin x \cos x \, dx = \sin^2 x \). From here, dividing through by 2 results in the integral equalling \( \frac{1}{2} \sin^2 x + C \). This method is particularly useful for handling products of sine and cosine functions, showcasing its versatility.

Trigonometric identities are equations involving trigonometric functions that are universally true. These identities simplify integrals and other mathematical problems by transforming complex expressions into more manageable forms.

In this problem, the identity \( \sin 2x = 2\sin x\cos x \) played a central role. This identity allowed us to rewrite the integral \( \int \sin x \cos x \, dx \) as \( \frac{1}{2} \int \sin 2x \, dx \). This reduction, in combination with known antiderivatives of trigonometric functions, simplifies calculations.

Further simplifications using \( \cos 2x = 1 - 2\sin^2 x \) maintain consistency with the final outcome of \( \frac{1}{2}\sin^2 x + C \). Recognizing and accurately applying these identities can drastically ease solving intricate integrals.

The substitution method is a fundamental integration technique often used to simplify integrals into more recognizable forms. It works by substituting a part of the integral with a new variable, turning the integral into something easier to handle.

For the integral \( \int \sin x \cos x \, dx \), we explored substitutions with \( u = \cos x \) and \( u = \sin x \). Each choice corresponds to different derivatives, with \( du = -\sin x \, dx \) for the first and \( du = \cos x \, dx \) for the second. Both transformations yielded expressions inline with \( \int u \, du \), leading to the final result of \( \frac{1}{2} \sin^2 x + C \).

This method reveals the flexibility of substitution in breaking down complex integrals and integrating intuition into mathematical manipulation for solutions.