Problem 6 - Applied Calculus: For Business, Economics, and the Social and Life Sciences

Question

$$ \int(\ln x)^{2} d x $$

Step-by-Step Solution

Answer

The integral is \( x((\ln x)^2 - 2 \ln x + 2) + C \).

Step 1: Use Integration by Parts Formula

Recall the formula for integration by parts: \[\int u dv = uv - \int v du\]Choose \(u = (\ln x)^2\) and \(dv = dx\).

Step 2: Differentiate and Integrate

Differentiate \(u\):\[du = 2 \ln(x) \cdot \frac{1}{x} dx = \frac{2 \ln(x)}{x} dx\]Integrate \(dv\):\[v = \int dx = x\]

Step 3: Apply the Integration by Parts

Substitute \(u, v, du,on\) back into the integration by parts formula:\[\int (\ln x)^2 dx = x(\ln x)^2 - \int x \cdot \frac{2 \ln x}{x} dx\]This simplifies to:\[\int (\ln x)^2 dx = x(\ln x)^2 - 2 \int \ln x \, dx\]

Step 4: Integrate \(\int \ln x \, dx\)

Use integration by parts again, letting \(u = \ln x\) and \(dv = dx\). Thus, \(du = \frac{1}{x} dx\) and \(v = x\):\[\int \ln x \, dx = x \ln x - \int x \cdot \frac{1}{x} dx = x \ln x - \int dx = x \ln x - x + C\]

Step 5: Combine the Results

Substitute back into the expression from Step 3:\[\int (\ln x)^2 dx = x(\ln x)^2 - 2(x \ln x - x)\]Simplify it:\[\int (\ln x)^2 dx = x(\ln x)^2 - 2x \ln x + 2x + C\]Combine like terms:\[\int (\ln x)^2 dx = x((\ln x)^2 - 2 \ln x + 2) + C\]

Key Concepts

calculus for business and economics

Calculus plays a crucial role in business and economics, helping in optimization, marginal analysis, and understanding changes over time. Among the integration techniques, definite integrals are particularly powerful:

• Optimization: Businesses use calculus to find the best conditions, like maximizing profit or minimizing costs. This generally involves finding where the derivative (rate of change) is zero.
• Marginal Analysis: The derivative of a cost function, revenue function, or profit function helps determine the additional cost or revenue for making one more unit.
• Consumer and Producer Surplus: Areas under demand and supply curves can be calculated using definite integrals, allowing businesses to assess economic welfare:

For example, if a business wants to forecast profits from sales, they might use definite integral to calculate the total profit over a particular period, given by: \[\text{Total Profit} = \int_a^b \text{Profit \, Function} \, dx\]Utilizing integration techniques, businesses can make informed decisions by quantifying various continuous changes effectively.