Problem 40 - Applied Calculus

Question

Find each indefinite integral. [Hint: Use some algebra first. \(\int \frac{(x+2)^{3}}{x} d x\)

Step-by-Step Solution

Answer

\(\int \frac{(x+2)^3}{x} \, dx = \frac{x^3}{3} + 3x^2 + 12x + 8\ln|x| + C\).

Step 1: Expand the Numerator

Start by expanding \((x+2)^3\) using the binomial theorem. The expansion is:\[(x+2)^3 = x^3 + 3x^2 \cdot 2 + 3x \cdot 2^2 + 2^3 = x^3 + 6x^2 + 12x + 8\]

Step 2: Divide Each Term by x

Now divide each term in the expanded form by \(x\):\[\frac{x^3}{x} + \frac{6x^2}{x} + \frac{12x}{x} + \frac{8}{x} = x^2 + 6x + 12 + \frac{8}{x}\]

Step 3: Integrate Term by Term

Integrate each term separately with respect to \(x\):1. Integrate \(x^2\): \(\int x^2 \, dx = \frac{x^3}{3} + C_1\)2. Integrate \(6x\): \(\int 6x \, dx = 3x^2 + C_2\)3. Integrate \(12\): \(\int 12 \, dx = 12x + C_3\)4. Integrate \(\frac{8}{x}\): \(\int \frac{8}{x} \, dx = 8\ln|x| + C_4\)

Step 4: Combine the Results

Combine the results from integrating each term to form the indefinite integral:\[\int \frac{(x+2)^3}{x} \, dx = \frac{x^3}{3} + 3x^2 + 12x + 8\ln|x| + C\]where \(C = C_1 + C_2 + C_3 + C_4\) is the constant of integration.

Key Concepts

Understanding the Binomial Theorem

The binomial theorem is a powerful algebraic tool used to expand expressions that are raised to a power, like \( (x + 2)^3 \). When applying the binomial theorem, you expand the expression into a series of terms based on the coefficients and exponents. In this theorem, each term takes the form of \( \binom{n}{k} a^{n-k} b^k \) where \( n \) is the exponent, and \( k \) varies from 0 to \( n \).

For \( (x + 2)^3 \), the expansion involves these steps:

• The first term is \( x^3 \) because \( \binom{3}{0} x^3 (2)^0 = x^3 \).
• Next, multiply \( x^2 \) by 3 and then by 2, to get \( 6x^2 \), from \( \binom{3}{1} x^2 (2)^1 \).
• The third term becomes \( 12x \), using \( \binom{3}{2} x^1 (2)^2 \).
• Finally, the constant term is \( 8 \) from \( \binom{3}{3} x^0 (2)^3 \).

This expansion turns a complex multiplication into simpler terms you can work with more easily.

Exploring Integration Techniques

Integration techniques are fundamental in calculus to find antiderivatives, also known as indefinite integrals. In this exercise, after expanding the expression using algebraic methods, each term is separately integrated. Every term of the polynomial and the rational expression \( \frac{8}{x} \) can be tackled with basic integration rules.

These steps include:

• Using the power rule: Integrate \( x^n \) by adding 1 to the exponent and dividing by the new exponent. So, \( \int x^2 \, dx \) becomes \( \frac{x^3}{3} \).
• Integrating constants: For a constant like 12, the integral with respect to \( x \) is simply multiplying the constant by \( x \).
• Integrating \( \frac{8}{x} \): Use the natural logarithm because the integral of \( \frac{1}{x} \) is \( \ln|x| \).

By integrating each term separately and then adding them together, you achieve the final result. Understanding these techniques simplifies complex integrals into manageable steps.

The Role of the Constant of Integration

The constant of integration, denoted by \( C \), plays a crucial role in indefinite integrals. Whenever you find the antiderivative of a function, you include \( C \) to represent the family of all possible solutions. This is important because differentiation of a constant results in zero, meaning many different antiderivatives can exist that differ by a constant.

In the final result of our integral, the constant \( C \) is the sum of all individual constants from each term's integration: \( C_1, \ C_2, \ C_3, \ \text{and} \ C_4 \).

• This combined constant ensures that the integral covers all possible shifts in the vertical direction on a graph.
• It represents the entire collection of functions whose derivative gives back the original function within the integral.

Neglecting this constant would mean missing out on many valid solutions of the given integral, making \( C \) indispensable in calculus.