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When dealing with expressions that involve powers, like \((x+1)^2\), it can be beneficial to expand these expressions using the binomial expansion. The binomial expansion is a way to express powers of binomials as a sum of terms. In this specific case, we apply the formula \((a+b)^2 = a^2 + 2ab + b^2\). This allows us to simplify \((x+1)^2\) into \(x^2 + 2x + 1\). Expanding expressions like this makes it easier to perform operations such as integration,because it transforms a single term with a power into multiple, more manageable, terms. By converting the binomial expression into a polynomial, we can comfortably proceed to the next step of integrating term by term.

The power rule of integration is a fundamental technique used to integrate terms of the form \(x^n\). This rule states that \(\int x^n \, dx = \frac{x^{n+1}}{n+1} + C\), as long as \(n eq -1\). Here, \(C\) is the constant of integration. The steps are simple:

• Increase the power of \(x\) by 1.
• Divide by the new power.
• Add the constant of integration \(C\).

In the example provided, we apply the power rule to each term of the polynomial separately:

• First Term: \(\int x^2 \, dx = \frac{x^3}{3} + C_1\)
• Second Term: \(\int 2x \, dx = x^2 + C_2\)
• Third Term: \(\int 1 \, dx = x + C_3\).

Each result includes a constant of integration, and these add up when we combine all the terms.

Polynomial integration involves integrating expressions like \(x^2 + 2x + 1\), which come from expanded binomial powers. Integrating these types of expressions becomes straightforward whenyou apply the power rule of integration term by term. Each element of the polynomial is integrated separately:

• Apply the power rule to the term \(x^2\).
• Use the same rule on \(2x\).
• For the constant term \(1\), recognize this integrates to \(x\) itself.

After performing integration on each segment of the polynomial, we combine them into a single expression: \(\int (x+1)^2 \ dx = \frac{x^3}{3} + x^2 + x + C\). This seamless merging of terms represents the indefinite integral of the original expression, where \(C\) stands for the unified constant of integration, acknowledging the accumulation of separate constants from each integration step.