91Ó°ÊÓ

The substitution method is a powerful technique used in calculus to simplify the process of finding integrals. It involves replacing the variable of integration with a new variable that might make the integral easier to evaluate.
In our exercise, we deal with the indefinite integral \( \int \frac{x^3}{\sqrt{1-x^2}} \ dx \). Here, it's helpful to use a trigonometric substitution. A suitable variable change can transform the complex expression into something more manageable.

• Identify an expression that is complicated, such as a compound function or a square root.
• Choose a substitution that simplifies this expression. In this case, \( x = \sin(\theta) \) is ideal because it transforms \( \sqrt{1-x^2} \) into \( \cos(\theta) \).
• Replace the original variable with the new one and remember to substitute \( dx \) as well (here, \( dx = \cos(\theta) \, d\theta \)).

This way, we transform the integral into a simpler form, making further analysis and integration possible.

Trigonometric substitution is especially powerful for integrals involving square roots of the form \( \sqrt{a^2 - x^2} \), \( \sqrt{a^2 + x^2} \), or \( \sqrt{x^2 - a^2} \). By substituting the variable \( x \) with a trigonometric function, we make use of Pythagorean identities.
In the exercise, we chose \( x = \sin(\theta) \) because \( 1-x^2 \) becomes \( \cos^2(\theta) \), simplifying the square root easily. This is suitable since:

• The identity \( 1 - \sin^2(\theta) = \cos^2(\theta) \) applies, reducing the square root in the denominator.
• \( \sin(\theta) \) and \( \cos(\theta) \) are well-understood functions with known derivative and integrals, making calculations straightforward.

With trigonometric substitution, the integral takes the form \( \int \sin^3(\theta) \, d\theta \). This form is manageable with basic integration techniques.

Integration by parts is a method derived from the product rule of differentiation and is used to integrate the product of two functions. The formula is given by:\[\int u \, dv = uv - \int v \, du.\]
In our problem, integration by parts is used indirectly when breaking down \( \int \sin(\theta) \cos^2(\theta) \, d\theta \). Instead, substituting \( u = \cos(\theta) \) converts this integral into a polynomial form \( \int -u^2 \, du \), which can be easily integrated using basic rules.
The idea is to decompose a complex integral into simpler parts:

• Choose \( u \) and \( dv \) from the integral such that \( dv \) is easily integrable.
• Calculate \( du \) by differentiating \( u \) and find \( v \) by integrating \( dv \).
• Apply the integration by parts formula to simplify the integral.

By applying these steps, we can handle integrals like \( \int u^2 \, du \) and solve them, making it easier to find the antiderivative of more complex expressions.