91Ó°ÊÓ

When you have a tricky integral to solve, like \( \int_{0}^{4} \sqrt{x(4-x)} \, dx \), trigonometric substitution can be a powerful tool. This technique involves simplifying the integrand by substituting a trigonometric function that suits the expression under the square root. In this case, the substitution \( x = 4 \sin^2(\theta) \) was chosen because it transforms the expression \( \sqrt{x(4-x)} \) into a much simpler form.
By letting \( x = 4 \sin^2(\theta) \), the differential changes to \( dx = 8 \sin(\theta)\cos(\theta) \, d\theta \). As a result, the integral becomes more manageable since \( \sqrt{x(4-x)} \) simplifies to \( 4 \sin(\theta)\cos(\theta) \). This substitution effectively turns a complex integral into one that is more straightforward and involves basic trigonometric functions.
Trigonometric substitution is particularly useful when the integrand includes expressions like \( a^2 - x^2 \), \( x^2 - a^2 \), or \( x^2 + a^2 \), where \( a \) is a constant. From here, the integration process is simplified, allowing you to utilize trigonometric identities and transformations.

When you apply substitution in integrals, it's crucial to update the limits of integration to correspond to the new variable. In our example, the original limits are from \( x = 0 \) to \( x = 4 \). After substituting \( x = 4 \sin^2(\theta) \), we need to express these limits in terms of \( \theta \).

• For \( x = 0 \), solving \( 4 \sin^2(\theta) = 0 \) gives \( \theta = 0 \).
• For \( x = 4 \), solving \( 4 \sin^2(\theta) = 4 \) gives \( \sin^2(\theta) = 1 \), leading to \( \theta = \frac{\pi}{2} \).

This means the integration now ranges from \( \theta = 0 \) to \( \theta = \frac{\pi}{2} \). This change of limits ensures that the integration remains accurate in the new variable context. The process highlights the importance of not only transforming the integrand but also adjusting the integration boundaries to align with the substitution.

Trigonometric identities play a vital role in simplifying integrals after substitution. They allow complex trigonometric expressions to be transformed into simpler forms. In this problem, the identity \( \sin^2(\theta)\cos^2(\theta) = \frac{1}{4}\sin^2(2\theta) \) was used to simplify the integrand.
By simplifying \( 32 \int_{0}^{\pi/2} \sin^2(\theta) \cos^2(\theta) \, d\theta \) to \( 8 \int_{0}^{\pi/2} \sin^2(2\theta) \, d\theta \), the integration process becomes much more straightforward. Additionally, the identity \( \sin^2(2\theta) = \frac{1}{2}(1 - \cos(4\theta)) \) further simplifies the problem and helps break down the integral into parts that are easier to evaluate.
Trigonometric identities like these are key strategies in reducing the complexity of integration tasks by rewriting components into expressions that lead to straightforward evaluation. These identities are foundational in calculus, offering tools to tackle even the most perplexing integrals.