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Chapter 9: Problem 4

Evaluate the following integrals: \(\int x(2 x-3)^{2} d x\)

Short Answer

Expert verified
The integral is \ \[ x^4 - 4x^3 + \frac{9}{2}x^2 + C. \]

Step by step solution

01

- Expand the Integrand

First, expand the binomial \( (2x - 3)^2 \) to make the integration easier. The expansion is \[ (2x - 3)^2 = (2x)^2 - 2 \times 2x \times 3 + 3^2 = 4x^2 - 12x + 9. \] This turns the integral into \[ \int x(4x^2 - 12x + 9) \, dx. \]
02

- Distribute the x

Distribute the x across each term inside the expanded polynomial: \[ x(4x^2 - 12x + 9) = 4x^3 - 12x^2 + 9x. \] This changes the integral to \[ \int (4x^3 - 12x^2 + 9x) \, dx. \]
03

- Integrate Each Term

Integrate each term separately using the power rule \( \int x^n dx = \frac{x^{n+1}}{n+1} \): \[ \int 4x^3 \, dx = \frac{4x^4}{4} = x^4, \] \[ \int -12x^2 \, dx = -12 \frac{x^3}{3} = -4x^3, \] \[ \int 9x \, dx = 9 \frac{x^2}{2} = \frac{9}{2}x^2. \]
04

- Combine the Results

Combine the results from each integrated term, including the constant of integration C: \[ x^4 - 4x^3 + \frac{9}{2}x^2 + C. \]

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Key Concepts

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

Integral Evaluation
The process of evaluating integrals involves finding the antiderivative of a function. This means we are looking for a function whose derivative equals the given function. In our exercise, we have the integral \(\int x(2x-3)^{2} \ d x\). The key steps include expanding the integrand, distributing terms, and then integrating each term separately.
Power Rule
The power rule for integration is an essential technique often used in integral calculus. It states that for any real number \(n eq -1\), the integral of \(x^n\) with respect to \(x\) is given by \[\int x^n \ dx = \frac{x^{n+1}}{n+1} + C\].
In our problem, we used the power rule to integrate each term after expanding and distributing.
Here are the power rule applications from our solution:
  • \(\int 4x^3 \ dx = \frac{4x^4}{4} = x^4\)
  • \(\int -12x^2 \ dx = -12 \frac{x^3}{3} = -4x^3\)
  • \(\int 9x \ dx = 9 \frac{x^2}{2} = \frac{9}{2}x^2\)
Binomial Expansion
Binomial expansion is used to expand expressions that are raised to a power. Here, we had to expand \((2x-3)^2\) to make the integral more manageable:

The general formula for binomial expansion is:
\((a-b)^n = \sum_{k=0}^n \binom{n}{k} a^{n-k}(-b)^k\)

For \((2x-3)^2\), using the formula:
\(a = 2x\) and \(-b = 3\)
Expanding we get:
\(4x^2 - 12x + 9\).
This simplifies the integral to:
\int x(4x^2 - 12x + 9) \ dx.
This step helps in simplifying the integrand for easier integration.
Definite and Indefinite Integrals
We have two main types of integrals: definite and indefinite.

  • Indefinite integrals represent a family of functions and include a constant of integration, \(C\). For instance:
    \(\int f(x) \ dx = F(x) + C\)
  • Definite integrals compute the area under a curve within a specified range, and do not include \(C\), for example: \(\int_{a}^{b} f(x) \ dx\)

In our example, we evaluated an indefinite integral,
resulting in:
\( x^4 - 4x^3 + \frac{9}{2} x^2 + C\).
This is because we integrated each term without specified limits. Remembering these differences is crucial for understanding and correctly solving integral problems.

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Most popular questions from this chapter

Approximate the following integrals by the midpoint rule; then, find the exact value by integration. Express your answers to five decimal places. \(\int_{0}^{1}\left(x-\frac{1}{2}\right)^{2} d x ; n=4\)

Consider \(\int_{0}^{2} f(x) d x\), where \(f(x)=\frac{1}{12} x^{4}+3 x^{2}\). (a) Make a rough sketch of the graph of \(f^{\prime \prime}(x)\) for \(0 \leq x \leq 2 .\) (b) Find a number \(A\) such that \(\left|f^{\prime \prime}(x)\right| \leq A\) for all \(x\) satisfying \(0 \leq x \leq 2\). (c) Obtain a bound on the error of using the midpoint rule with \(n=10\) to approximate the definite integral. (d) The exact value of the definite integral (to four decimal places) is \(8.5333\), and the midpoint rule with \(n=10\) gives \(8.5089 .\) What is the error for the midpoint approximation? Does this error satisfy the bound obtained in part (c)? (e) Redo part (c) with the number of intervals doubled to \(n-20\). Is the bound on the error halved? Quartered?

Approximate the following integrals by the midpoint rule; then, find the exact value by integration. Express your answers to five decimal places. \(\int_{-1}^{1} e^{2 x} d x ; n=2,4\)

Approximate the following integrals by the midpoint rule, the frapezoidal rule, and Simpson's rule. Then, find the exact value by integration. Express your answers to five decimal places. \(\int_{10}^{20} \frac{\ln x}{x} d x ; n=5\)

To determine the amount of water flowing down a certain 100-yard-wide river, engineers need to know the area of a vertical cross section of the river. Measurements of the depth of the river were made every 20 yards from one bank to the other. The readings in fathoms were \(0,1,2,3,1,0 .\) (One fathom equals 2 yards.) Use the trapezoidal rule to estimate the area of the cross section.
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