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Chapter 5: Problem 39

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

Short Answer

Expert verified
The integral is \( \frac{x^3}{3} - 3x^2 + 12x - 8\ln|x| + C \).

Step by step solution

01

Simplify the Integrand

First, perform algebraic division to simplify the integrand. Divide \((x-2)^3\) by \(x\):\[(x-2)^3 = (x^3 - 6x^2 + 12x - 8)\].By dividing each term by \(x\), we get: \[\frac{x^3}{x} - \frac{6x^2}{x} + \frac{12x}{x} - \frac{8}{x} = x^2 - 6x + 12 - \frac{8}{x}\].The integral becomes:\[\int (x^2 - 6x + 12 - \frac{8}{x}) \, dx\].
02

Integrate Term by Term

Now integrate each term separately:\[\int x^2 \, dx = \frac{x^3}{3}\], \[\int (-6x) \, dx = -3x^2\], \[\int 12 \, dx = 12x\], \[\int (-\frac{8}{x}) \, dx = -8\ln|x|\].
03

Combine the Integrals

Combine the results of each individual integral:\[\frac{x^3}{3} - 3x^2 + 12x - 8\ln|x| + C\],where \(C\) is the constant of integration.

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Key Concepts

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

Algebraic Simplification
Before diving into integrating complex expressions, simplifying the integrand through algebraic manipulation can make the solution much easier. To handle an expression such as \( \frac{(x-2)^3}{x} \), it's useful to simplify by expanding the numerator fully. This way, each term can be divided by the denominator individually.For example:
  • Expand \((x-2)^3\) to get \(x^3 - 6x^2 + 12x - 8\).
  • Divide each term by \(x\), resulting in \(x^2 - 6x + 12 - \frac{8}{x}\).
This simplification turns a seemingly complex fraction into simpler polynomial terms, thereby paving the path for straightforward integration. By taking the time to simplify algebraically, it becomes easier to apply standard integration rules.
Integration Techniques
Once the integrand has been simplified, the next step is to integrate it term by term. Each part of the polynomial can be tackled using basic integration rules.Here are some useful techniques:
  • For a power of \(x\), such as \(x^n\), use the formula \(\int x^n \, dx = \frac{x^{n+1}}{n+1}\), applicable for non-zero \(n\).
  • For constants multiplied by \(x\), such as \(-6x\), simply apply \(\int -6x \, dx = -3x^2\).
  • For constant terms \(c\), the integration is straightforward: \(\int 12 \, dx = 12x\).
  • For the term \(-\frac{8}{x}\), recognize it as a natural logarithmic function \(\int -\frac{8}{x} \, dx = -8\ln|x|\).
These techniques simplify integration into manageable steps, helping to systematically work through different types of terms in the expression.
Calculus Problem Solving
In calculus problem solving, understanding each step in the integration process is crucial for finding the final solution and verifying its correctness. After integrating each term in the integrand separately, it's essential to combine them properly to form the complete solution.Steps to finalize your solution:
  • Integrate each term separately: Start from the highest power of \(x\) and proceed stepwise to easier terms, including constants.
  • Combine the integrated terms: In our example, this yields: \( \frac{x^3}{3} - 3x^2 + 12x - 8\ln|x| \).
  • Don't forget the constant of integration \(C\): This is a crucial part when dealing with indefinite integrals as it represents the family of antiderivatives.
By following these steps, students develop a structured approach to calculus problems, which helps in achieving the correct solution efficiently. Each solved integration problem strengthens problem-solving skills and comprehension of calculus concepts.

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Most popular questions from this chapter

BIOMEDICAL: Poiseuille's Law According to Poiseuille's law, the speed of blood in a blood vessel is given by \(V=\frac{p}{4 L v}\left(R^{2}-r^{2}\right)\) where \(R\) is the radius of the blood vessel, \(r\) is the distance of the blood from the center of the blood vessel, and \(p, L,\) and \(v\) are constants determined by the pressure and viscosity of the blood and the length of the vessel. The total blood flow is then given by $$ \left(\begin{array}{c} \text { Total } \\ \text { blood flow } \end{array}\right)=\int_{0}^{R} 2 \pi \frac{p}{4 L v}\left(R^{2}-r^{2}\right) r d r $$ Find the total blood flow by finding this integral \((p, L, v,\) and \(R\) are constants)

Find a formula for \(\int_{a} c d x .\) [Hint: No calculation necessary-just think of a graph.]

A real estate office is selling condominiums at the rate of \(100 e^{-x / 4}\) per week after \(x\) weeks. How many condominiums will be sold during the first 8 weeks?

How will \(\int_{a}^{b} f(x) d x\) and \(\int_{b}^{a} f(x) d x\) differ? [Hint: Assume that they can be evaluated by the Fundamental Theorem of Integral Calculus, and think how they will differ at the "evaluate and subtract" step.

For each definite integral: a. Evaluate it "by hand." b. Check your answer by using a graphing calculator. $$ \int_{2}^{3} \frac{d x}{1-x} $$
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