Problem 22 - Applied Calculus

Question

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

Step-by-Step Solution

Answer

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

Step 1: Identify the Integral Type

The given integral \( \int (x^2 + 3)^2 \, dx \) is a polynomial raised to a power. This can be expanded or solved using substitution methods.

Step 2: Expand the Polynomial

Before integrating, expand the polynomial inside the integral: \((x^2 + 3)^2 = x^4 + 6x^2 + 9\). The integral now becomes \( \int (x^4 + 6x^2 + 9) \, dx \).

Step 3: Integrate Term by Term

Integrate each term separately:1. \( \int x^4 \, dx = \frac{x^5}{5} \)2. \( \int 6x^2 \, dx = 6 \cdot \frac{x^3}{3} = 2x^3 \)3. \( \int 9 \, dx = 9x \).Combine these results: \( \frac{x^5}{5} + 2x^3 + 9x + C \).

Step 4: Differentiate the Result

Differentiate \( \frac{x^5}{5} + 2x^3 + 9x + C \) to confirm the integral:1. \( \frac{d}{dx} \left( \frac{x^5}{5} \right) = x^4 \)2. \( \frac{d}{dx} (2x^3) = 6x^2 \)3. \( \frac{d}{dx} (9x) = 9 \).Combine these: the derivative is \( x^4 + 6x^2 + 9 \), which matches the expanded expression inside the integral.

Key Concepts

Polynomial Expansion

Polynomial expansion is a powerful tool that simplifies complex expressions by expressing them as sums of simpler terms. Consider the polynomial expression \( \int (x^2 + 3)^2 \, dx \).To make it easier to integrate, we can expand this polynomial using distributive multiplication.

The expression \( (x^2 + 3)^2 \)effectively means we are multiplying \( (x^2 + 3) \)by itself.

• First, multiply \( x^2 \)by both terms of the binomial:
  • \( x^2 \times x^2 = x^4 \)
  • \( x^2 \times 3 = 3x^2 \)
• Then, do the same for the number 3:
  • \( 3 \times x^2 = 3x^2 \)
  • \( 3 \times 3 = 9 \)
• Finally, combine all these terms:\( x^4 + 3x^2 + 3x^2 + 9 \), simplify to get \( x^4 + 6x^2 + 9 \).

This expanded form allows us to integrate each term individually, which is much simpler to handle.

Differentiation

Differentiation is the process of finding the derivative of a function. A derivative represents how a function changes as its input changes. After integrating in our example, the final integrated expression is \( \frac{x^5}{5} + 2x^3 + 9x + C \).

To verify the integration, we'll differentiate this expression to ensure it yields the original polynomial \( (x^2 + 3)^2 \) when expanded, which is \( x^4 + 6x^2 + 9 \).

• Differentiate \( \frac{x^5}{5} \),giving \( x^4 \)because we multiply by the original exponent and decrease it by 1.
• For \( 2x^3 \),differentiation yields \( 6x^2 \).Multiply the coefficient by 3 and decrease the power.
• The term \( 9x \)differentiates to 9, as the derivative of \( x \)is 1.

Combining these, the derivative is \( x^4 + 6x^2 + 9 \), confirming our integration was correct. Differentiation acts as a check here, ensuring the operations performed were accurate.

Substitution Methods

Substitution is an alternative method to solve integrals, especially when dealing with functions inside another function, like \( (x^2 + 3)^2 \).This method simplifies an equation before further manipulative steps like integration.

In some cases, instead of expanding, we can use a substitution such as \( u = x^2 + 3 \),then \( du = 2x \, dx \).However, if the integral's form is more closely aligned with polynomial terms as in our example, expansion might be more straightforward.

Using substitution involves:

• Identifying an inner function to represent as \( u \).
• Calculating \( du \),which is crucial for expressing the integrand solely in terms of new variables.
• Substituting \( u \)and \( du \)back into the integral simplifies it, ideally leading to easily integrable terms.
• Finally, revert to the original variable for the integral's solution.

In our example, however, we found expansion more straightforward and efficient due to the polynomial's simple structure. It demonstrates the importance of choosing the right method based on the problem at hand.