Problem 41 - Calculus and its applications

Question

Find each integral. $$ \int(3 x+2)^{2} d x \text { (Hint: Expand first.) } $$

Step-by-Step Solution

Answer

The integral is \(3x^3 + 6x^2 + 4x + C\).

Step 1: Expand the Expression

The hint suggests expanding the expression inside the integral first. Let's expand \((3x + 2)^2\):\[(3x + 2)^2 = (3x + 2)(3x + 2) = 3x imes 3x + 3x imes 2 + 2 imes 3x + 2 imes 2\]Simplifying this gives:\[= 9x^2 + 6x + 6x + 4 = 9x^2 + 12x + 4\]

Step 2: Set up the Integral of Expanded Expression

Set up the integral with the expanded expression:\[\int(3x+2)^2 dx = \int (9x^2 + 12x + 4) dx\]

Step 3: Integrate Term by Term

Integrate each term separately:1. For \(9x^2\), the integral is:\[\int 9x^2 \, dx = 9 \cdot \frac{x^{3}}{3} = 3x^3\]2. For \(12x\), the integral is:\[\int 12x \, dx = 12 \cdot \frac{x^2}{2} = 6x^2\]3. For the constant \(4\), the integral is:\[\int 4 \, dx = 4x\]

Step 4: Combine the Integrals and Add Constant of Integration

Combine the results from the previous step, remembering to include the constant of integration \(C\):\[\int (9x^2 + 12x + 4) \, dx = 3x^3 + 6x^2 + 4x + C\]

Key Concepts

Polynomial Expansion

When dealing with integral calculus, sometimes the best starting point is simplifying the expression at hand. A common technique is polynomial expansion, especially when the function you're dealing with is in a compact form. For example, if you have an expression like \((3x + 2)^2\), it can be expanded using the distributive property. This involves multiplying out the binomials as follows:

• First, calculate \((3x + 2)(3x + 2)\)
• Multiply term by term: \(3x \times 3x = 9x^2\)
• \(3x \times 2 = 6x\)
• \(2 \times 3x = 6x\)
• \(2 \times 2 = 4\)

After accounting for all these products, you combine like terms: \(9x^2 + 6x + 6x + 4\) simplifies to \(9x^2 + 12x + 4\). By expanding first, the integration process becomes easier because each term can be handled individually.

Indefinite Integral

Once you have expanded the polynomial, the next step in the process is to find its indefinite integral. Indefinite integrals, unlike definite integrals, do not have specified limits and hence their result is a general function plus a constant. The process involves integrating each term separately:

• The integral of \(9x^2\) is computed as \(\int 9x^2 \, dx = 9 \cdot \frac{x^3}{3} = 3x^3\).
• For \(12x\), integrating gives \(\int 12x \, dx = 12 \cdot \frac{x^2}{2} = 6x^2\).
• The constant term \(4\) is integrated as \(\int 4 \, dx = 4x\).

Breaking it down like this allows you to address each term individually, thereby simplifying the computation. The indefinite integral is essentially the sum of these individual integrals of monomial terms.

Constant of Integration

In the world of calculus, the constant of integration is a key concept when solving indefinite integrals. It emerges from the nature of differentiation being a unique process while integration allows for a family of functions. After integrating the expression and obtaining a formula that describes an antiderivative, we add a constant, \(C\), to represent all possible vertical shifts of the antiderivative function. Remember:

• Every indefinite integral yields an antiderivative plus a constant \(C\).
• This constant accounts for the fact that the derivative of a constant is zero.
• Without \(C\), the solution would imply only one possible function derived from the integral, ignoring the whole family of solutions that differ by a constant.

In our exercise, the complete integral result is \(3x^3 + 6x^2 + 4x + C\), where \(C\) is crucial for capturing all potential solutions to the integral problem.