91Ó°ÊÓ

The substitution method is a technique often used to make integration problems easier to solve. It involves changing variables to simplify the integral.
For example, in the problem \( \int \frac{x^{3}}{\sqrt{1-x^{4}}} \, dx \), we recognize a pattern that suggests substitution might simplify our work.
We see \( x^4 \) in the square root, which points towards a natural substitution. By letting \( u = x^4 \), the differential \( du = 4x^3 \, dx \) is derived, which directly corresponds to the terms found within the integral.
Rewriting the integral in terms of \( u \) involves expressing all other variables and terms in the integral using \( u \), resulting in \( \int \frac{1}{\sqrt{1-u}} \cdot \frac{1}{4} \, du \). This converted form often reveals a simpler path to integrating.

When dealing with integrals, it's essential to distinguish between definite and indefinite integrals.
Indefinite integrals involve finding the antiderivative of a function and include a constant of integration, denoted by \( C \). This is what we worked with in the given exercise.
In the original problem, \( \int \frac{x^{3}}{\sqrt{1-x^{4}}} \, dx \), we ended up finding the antiderivative \( \frac{1}{2} (1-x^4)^{\frac{1}{2}} + C \). This solution is indefinite because it represents a family of functions, each differing by an unknown constant \( C \).
Definite integrals, on the other hand, calculate the actual area under a curve and have specific limits of integration, providing a numerical answer rather than an algebraic expression.

The power rule for integration is a fundamental technique used to integrate functions that are polynomials or can be restructured into polynomial forms. The rule states that the integral \( \int x^n \, dx \) equals \( \frac{x^{n+1}}{n+1} + C \), provided \( n eq -1 \).
In our exercise, after substitution, the integral transforms into \( \int (1-u)^{-\frac{1}{2}} \, du \). This change renders the expression eligible for applying the power rule.
By increasing the exponent by 1 and dividing by the new exponent, we find the antiderivative: \( \frac{(1-u)^{\frac{1}{2}}}{\frac{1}{2}} = 2(1-u)^{\frac{1}{2}} \).
Finally, multiplying by the initial factor (\( \frac{1}{4} \) in this example, derived from our substitution), we reach the end result for this part of the integration process.