91Ó°ÊÓ

To solve the integral \( \int \frac{x}{\sqrt{9-x^{2}}} \, dx \) using integration techniques, we employ a method known as **trigonometric substitution**. This technique is particularly useful when you spot expressions in the form of \( \sqrt{a^2 - x^2} \). The square root suggests the use of trigonometric functions because of their relationship with the Pythagorean identity.So why do we use the substitution \( x = a \sin \theta \)?- **Simplification:** It transforms needs like \( \sqrt{9-x^2} \) to simpler trigonometric forms, making them easier to work with.- **Fitting Forms:** We often see \( a^2 \sin^2 \theta + a^2 \cos^2 \theta = a^2 \), aligning perfectly with identities.- **Differential Transformation:** The differential \( dx \) is rewritten. For instance, when \( x = 3 \sin \theta \), we get \( dx = 3 \cos \theta \, d\theta \), transforming the integral entirely.Once substitution is complete, the integral changes form entirely into a trigonometric integral, which can be solved using basic calculus rules like substitution back into x at the end.

In calculus, integrals come in two types: **definite** and **indefinite**.- **Indefinite Integrals:** These integrals are represented with the general form \( \int f(x) \, dx \) and include a constant \( C \). The constant arises because differentiation can reduce different functions to the same derivative (e.g., \( x+C \)). Thus, solving an indefinite integral leads to a family of possible antiderivatives.- **Definite Integrals:** Unlike indefinite, definite integrals have limits. They calculate the net area under a curve and are noted as \( \int_{a}^{b} f(x) \, dx \). This type doesn't include the constant \( C \) because the result is a specific numerical value.In our exercise, since the problem lacks limits, it's an indefinite integral. The solution provides a general function that represents a family of functions, each shifted vertically by some constant. The integral \( \int \sin \theta \, d\theta \) results in \( -\cos \theta + C \) in trigonometric terms, and we eventually back-substitute to express it in terms of \( x \).

Integrals involving square roots, such as \( \sqrt{9-x^2} \), are frequent yet challenging in calculus. This form specifically suggests using trigonometric substitution due to its resemblance to the Pythagorean identity: \( a^2 = x^2 + y^2 \).- **Form Recognition:** When you see \( \sqrt{a^2-x^2} \), it's wise to think about the substitution \( x = a \sin \theta \). This step will lead to simplifications: \( \sqrt{9-x^2} \) becomes \( 3\cos\theta \).- **Rationalizing the Expression:** Through this substitution, the root transforms to simpler trigonometric identities, easing the integration process.- **Effect on Differential:** The substitution not only simplifies the integrand but also transforms the differential, making the integral approachable.The end goal is to solve the transformed integral and back-substitute to return to the original variable. Importantly, such substitutions are best when turning complex root-based expressions into tractable trigonometric forms.