Problem 17 - Applied Calculus

Question

Find the integrals in problems. Check your answers by differentiation. $$ \int\left(x^{2}+3\right)^{2} d x $$

Step-by-Step Solution

Answer

The integral is \(\frac{x^5}{5} + 2x^3 + 9x + C\).

Step 1: Expand the integrand

First, we need to expand \((x^2 + 3)^2\). Use the formula \((a+b)^2 = a^2 + 2ab + b^2\). Here, \(a = x^2\) and \(b = 3\). Thus, the expansion is:\[(x^2 + 3)^2 = (x^2)^2 + 2(x^2)(3) + 3^2 = x^4 + 6x^2 + 9\]

Step 2: Integrate each term

Now, integrate each term separately:- For \(x^4\), the integral is \(\frac{x^5}{5}\).- For \(6x^2\), the integral is \(6 \cdot \frac{x^3}{3} = 2x^3\).- For \(9\), the integral is \(9x\).Therefore, the integral of \((x^2 + 3)^2\) is:\[\int (x^2 + 3)^2 \, dx = \frac{x^5}{5} + 2x^3 + 9x + C\]where \(C\) is the constant of integration.

Step 3: Differentiate the result to verify

Differentiate the result we obtained to check our work. Start by differentiating each term:- The derivative of \(\frac{x^5}{5}\) is \(x^4\).- The derivative of \(2x^3\) is \(6x^2\).- The derivative of \(9x\) is \(9\).- The derivative of \(C\) is \(0\), which we ignore.Thus, the derivative of our integral is:\[x^4 + 6x^2 + 9\]This matches the expanded integrand from Step 1, verifying our solution is correct.

Key Concepts

Polynomial Expansion

Polynomial expansion is a fundamental process in algebra that provides a simpler form of an expression by eliminating the parenthesis. It is particularly useful when integrating complex polynomials. When dealing with the expression \((x^2 + 3)^2\), the formula

• \((a+b)^2 = a^2 + 2ab + b^2\)

can be applied. The formula helps to systematically break down the expression into simpler polynomial terms that are easily integrable or differentiable.
For our specific case: if \(a = x^2\) and \(b = 3\), then:

• \((x^2)^2 = x^4\)
• \(2(x^2)(3) = 6x^2\)
• \(3^2 = 9\)

Combining these expanded results, we rewrite the polynomial as \(x^4 + 6x^2 + 9\). Thus, polynomial expansion simplifies the integration process by reducing it to the sum of integrals of each term.

Definite and Indefinite Integrals

Integral calculus is divided into two main types: definite and indefinite integrals. This exercise focuses on indefinite integrals, which determine a family of functions and include a constant of integration. When integrating polynomial expressions like \((x^2 + 3)^2\), it usually involves several steps:

• Expanding the polynomial
• Integrating each expanded term separately

Given the expanded polynomial \(x^4 + 6x^2 + 9\), you integrate each part:

• The integral of \(x^4\) is \(\frac{x^5}{5}\)
• The integral of \(6x^2\) is \(2x^3\)
• The integral of \(9\) is \(9x\)

Indefinite integration involves adding these results together, plus the constant of integration, \(C\), giving the solution: \(\frac{x^5}{5} + 2x^3 + 9x + C\). This represents a family of curves shifted vertically by \(C\), where calculus establishes a complete solution by considering boundary conditions in the case of definite integrals.

Verification by Differentiation

Verification by differentiation is a crucial step in confirming that an integral is computed correctly. After integrating \((x^2 + 3)^2\) to get \(\frac{x^5}{5} + 2x^3 + 9x + C\), the correctness of this result can be checked via differentiation. Differentiating an integral should return to the original polynomial function.
Here's the process for each term:

• The derivative of \(\frac{x^5}{5}\) results in \(x^4\)
• The derivative of \(2x^3\) results in \(6x^2\)
• The derivative of \(9x\) results in \(9\)
• The derivative of \(C\), being a constant, results in \(0\)

Performing these steps, the differentiation reconstructs the original expanded polynomial \(x^4 + 6x^2 + 9\). This confirms the method's accuracy, ensuring that the integration was done properly. Verification by differentiation is a valuable tool in calculus, assuring students that their solutions are correct.