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Chapter 6: Problem 6

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

Short Answer

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

Step by step solution

01

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 \(dv = dx\).
02

Differentiate and Integrate

Differentiate \(u\):\[du = 2 \ln(x) \cdot \frac{1}{x} dx = \frac{2 \ln(x)}{x} dx\]Integrate \(dv\):\[v = \int dx = x\]
03

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:\[\int (\ln x)^2 dx = x(\ln x)^2 - 2 \int \ln x \, dx\]
04

Integrate \(\int \ln x \, dx\)

Use integration by parts again, letting \(u = \ln x\) and \(dv = dx\). Thus, \(du = \frac{1}{x} dx\) and \(v = x\):\[\int \ln x \, dx = x \ln x - \int x \cdot \frac{1}{x} dx = x \ln x - \int dx = x \ln x - x + C\]
05

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:\[\int (\ln x)^2 dx = x((\ln x)^2 - 2 \ln x + 2) + C\]

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

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.

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Most popular questions from this chapter

Approximate the given integral and estimate the error with the specified number of subintervals using: (a) The trapezoidal rule. (b) Simpson's rule. $$ \int_{1}^{2} \frac{e^{x}}{x} d x ; n=10 $$

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$$ \int_{0}^{1} \frac{x+2}{e^{3 x}} d x $$

Either evaluate the given improper integral or show that it diverges. $$ \int_{0}^{+\infty}(1+2 x)^{-3 / 2} d x $$

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