91Ó°ÊÓ

Binomial Expansion is a powerful algebraic method used to expand expressions raised to a power. It follows specific patterns based on the binomial theorem. In the context of our integral, we were tasked with expanding \((2 + y^2)^2\).
For this, we used the formula:

• \((a + b)^2 = a^2 + 2ab + b^2\)

In our case, \(a = 2\) and \(b = y^2\), so applying the formula:

• \(2^2 = 4\)
• \(2 \times 2 \times y^2 = 4y^2\)
• \((y^2)^2 = y^4\)

Therefore, the expanded form is \(4 + 4y^2 + y^4\). This expansion simplifies the process of integrating polynomial expressions because each term can be dealt with separately.

Differentiation is the process by which we find the rate at which a function is changing at any given point. It's essentially the reverse process of integration. When we differentiate a polynomial, we systematically reduce the powers of each term.
For example, when differentiating our solution to check correctness:

• The derivative of \(4y\) gives us \(4\)
• The derivative of \(\frac{4}{3}y^3\) yields \(4y^2\)
• The derivative of \(\frac{1}{5}y^5\) gives \(y^4\)

Adding these derivatives results in the original expanded expression \(4 + 4y^2 + y^4\). This step acts as a verification showing that our integration was performed correctly, as the differentiation returns us to the initial integrand.

Indefinite Integrals represent the family of all antiderivatives of a function. The process of integration is essentially finding a function whose derivative gives us the original function. When dealing with indefinite integrals, a constant \(C\) is usually added to represent all the possible vertical shifts of the antiderivative.
In this exercise, after expanding and integrating term by term:

• The integral of \(4\) results in \(4y\)
• The integral of \(4y^2\) is \(\frac{4}{3}y^3\)
• The integral of \(y^4\) results in \(\frac{1}{5}y^5\)

We combine these to arrive at \(4y + \frac{4}{3}y^3 + \frac{1}{5}y^5 + C\). This expression represents an indefinite integral and covers all possible solutions by accounting for an unknown constant \(C\), which is crucial when solving real-world problems, as it allows for more variables in initial conditions.