Problem 26 - Applied Calculus

Question

Find each indefinite integral. \(\int(x+2)^{2} d x\)

Step-by-Step Solution

Answer

\( \int(x+2)^2 \, dx = \frac{x^3}{3} + 2x^2 + 4x + C.\)

Step 1: Expand the Integrand

First, we need to expand the expression \(x+2\)^{2}\ before integrating. To expand, we use the binomial theorem or simply multiply: \[(x+2)^{2} = x^{2} + 4x + 4.\] Thus, the integral can be rewritten as \int (x^2 + 4x + 4) \, dx\.

Step 2: Integrate Each Term

Next, we integrate each term separately. To do this, apply the power rule for integration: 1. \ \int x^2 \, dx = \frac{x^3}{3},\ 2. \ \int 4x \, dx = 4 \cdot \frac{x^2}{2} = 2x^2,\ 3. \ \int 4 \, dx = 4x.\Now, combine these to get the complete integral: \[\frac{x^3}{3} + 2x^2 + 4x.\]

Step 3: Add the Constant of Integration

Finally, add the constant of integration \" + C\" to the result, as we are finding an indefinite integral: \[\int (x^2 + 4x + 4) \, dx = \frac{x^3}{3} + 2x^2 + 4x + C.\]

Key Concepts

Binomial Theorem

The binomial theorem is a powerful tool that helps expand expressions raised to a power. Here, we used it to expand \((x + 2)^2\). This theorem provides a way to break down the power of a binomial into a sum of terms. Each term consists of a combination of the binomial's components.

In our example, \((x + 2)^2\), we apply the formula:\[(a + b)^n = \sum_{k=0}^{n} \binom{n}{k} a^{n-k} b^k.\]In this formula, \(a = x\) and \(b = 2\). The exponent \(n\) is 2. Expanding it gives the expression:

• \(x^2\)
• \(4x\)/(from combination of \(x\) and \(2\))
• \(+ 4\)/(constant term \(2^2\))

Using the binomial theorem in this way simplifies the expression, making integration easier in later steps.

Power Rule for Integration

The power rule is a basic but essential part of integration. It states that when integrating a term \(x^n\), the result is given by\(\frac{x^{n+1}}{n+1}\), provided \(n eq -1\).

Applying this rule simplifies the integration process. In our example with the expanded polynomial \(x^2 + 4x + 4\):

• For \(x^2\), the power increases to 3, giving us \(\frac{x^3}{3}\).
• For the \(4x\) term, the new power is 2, resulting in \(2x^2\).
• The constant \(4\), being multiplied by \(x\), results in \(4x\)

Once each term is integrated individually, combining them gives us the complete antiderivative. This illustrates how applying the power rule step-by-step can efficiently evaluate an indefinite integral.

Constant of Integration

When working with indefinite integrals, it's crucial to remember the constant of integration, often symbolized as \(C\). This "\(+ C\)" represents an unknown constant that accounts for all possible antiderivatives of a function.

Antiderivatives are not unique. Without specifically defining a constant term, each indefinite integral could potentially represent a family of curves. The constant of integration provides the essential flexibility to cover all potential functions that, when differentiated, lead to the original integrable expression.
For the example integral given as \(\int (x^2 + 4x + 4) \, dx\), adding \(+ C\) is necessary. It confirms you've covered all solutions possible, leading to \(\frac{x^3}{3} + 2x^2 + 4x + C\) as the complete antiderivative.