91Ó°ÊÓ

Differentiation is a key concept in integral calculus and mathematics as a whole. It involves computing the derivative of a function, which is a measure of how a function's value changes as its input changes.
In our exercise, differentiation is used as a verification step to check the correctness of the integrated solution. The derived formula should equal the original integrand if the integration was performed correctly.
When differentiating the expression \(2x - \frac{x^2}{2} + \frac{2x^3}{3} - \frac{x^4}{4}\), we obtain \(2 - x + 2x^2 - x^3\), exactly matching the expanded integrand. This shows that our previous calculations were correct. Differentiation thus acts as a proof or check for the integration process.

Polynomial integration is a process where we find the antiderivative or integral of a polynomial function. This involves applying the power of integrals separately to each term of the polynomial.
When integrating polynomials, one must increase the exponent by 1 and then divide by that new exponent. For example, the integral of \( x^n \) is \( \frac{x^{n+1}}{n+1} \).
In our original problem, we expanded the polynomial \( (1 + x^2)(2 - x) \) to get \(2 - x + 2x^2 - x^3\). Then, we integrated each term:

• \(\int 2 \, dx \to 2x\)
• \(\int x \, dx \to \frac{x^2}{2}\)
• \(\int 2x^2 \, dx \to \frac{2x^3}{3}\)
• \(\int x^3 \, dx \to \frac{x^4}{4}\)

This step-by-step integration of each term helps in building up the complete integrated function.

Whenever we perform an indefinite integral, there is always a constant of integration \(C\) involved. This is because an antiderivative is not unique; adding any constant \(C\) would still satisfy the differentiation requirement.
For example, when integrating \(\int 2 \, dx = 2x + C\), here \(C\) represents any constant value. Any function of the form \(2x + C\) will have a derivative of \(2\), thus showing how the constant doesn't affect the derivative.
The constant of integration is crucial in solving initial value problems, where you need to find a specific function that satisfies a particular condition or boundary value.
In our exercise, after integrating \(2x - \frac{x^2}{2} + \frac{2x^3}{3} - \frac{x^4}{4}\), we include the constant \(C\) to acknowledge the indefinite nature of the integral.