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Chapter 4: Problem 24

Evaluate the integral. $$ \int_{1}^{2} \frac{3 x^{4}-2 x^{2}+1}{2 x^{2}} d x $$

Short Answer

Expert verified
The short answer based on the given step-by-step solution is: To evaluate the integral, first simplify the integrand: \(\frac{3}{2}x^2 - 1 + \frac{1}{2x^2}\). Next, find the antiderivative: \(\frac{1}{2}x^3 - x + \frac{-1}{2x} + C\). Finally, evaluate the definite integral using the Fundamental Theorem of Calculus: \(\int_{1}^{2} \frac{3 x^{4}-2 x^{2}+1}{2 x^{2}} d x = \frac{7}{4}\).

Step by step solution

01

Simplify the integrand

First, we simplify the integrand by dividing each term in the numerator by \(2x^2\): \[\frac{3x^4 - 2x^2 + 1}{2x^2} = \frac{3x^4}{2x^2} - \frac{2x^2}{2x^2} + \frac{1}{2x^2} = \frac{3}{2}x^2 - 1 + \frac{1}{2x^2}\] Now, the simplified integrand is \(\frac{3}{2}x^2 - 1 + \frac{1}{2x^2}\).
02

Find the antiderivative

Now, we find the antiderivative of the simplified integrand: \[A(x) = \int{\left(\frac{3}{2}x^2 - 1 + \frac{1}{2x^2}\right)dx}\] To integrate this, we break it up into three separate integrals and integrate term by term: \[A(x) = \int{\frac{3}{2}x^2dx} - \int{1dx} + \int{\frac{1}{2x^2}dx}\] Now, we can integrate each term: \[A(x) = \frac{1}{2}x^3 - x + \frac{-1}{2x} + C\]
03

Evaluate the definite integral

Now, we evaluate the definite integral using the Fundamental Theorem of Calculus: \[\int_{1}^{2} \frac{3 x^{4}-2 x^{2}+1}{2 x^{2}} d x = A(2) - A(1)\] Substitute the values of 2 and 1 into the antiderivative and find the difference: \[\begin{aligned} A(2) - A(1) &= \left(\frac{1}{2}(2)^3 - (2) + \frac{-1}{2(2)}\right) - \left(\frac{1}{2}(1)^3 - (1) + \frac{-1}{2(1)}\right) \\ &= \left(4 - 2 - \frac{1}{4}\right) - \left(\frac{1}{2} - 1 + \frac{-1}{2}\right) \\ &= (2 - \frac{1}{4}) - (-\frac{1}{2}) \\ &= \frac{7}{4} \end{aligned}\] So, the value of the integral is: \[\int_{1}^{2} \frac{3 x^{4}-2 x^{2}+1}{2 x^{2}} d x = \frac{7}{4}\]

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Key Concepts

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

Antiderivative
An antiderivative is a function whose derivative is the original function we started with. In simpler terms, when you take the derivative of an antiderivative, you get the integrand back. Finding an antiderivative involves reverse-engineering differentiation.
For example, consider the function in the integrand \[ \frac{3}{2}x^2 - 1 + \frac{1}{2x^2}. \]
To find its antiderivative, we integrate each term:
  • For \( \frac{3}{2}x^2 \), its antiderivative is \( \frac{1}{2}x^3 \) because the derivative of \( \frac{1}{2}x^3 \) is \( \frac{3}{2}x^2 \).
  • The antiderivative of \(-1\) is \(-x\), since the derivative of \(-x\) is \(-1\).
  • Lastly, \( \frac{1}{2x^2} \) can be rewritten as \( \frac{1}{2}x^{-2} \), whose antiderivative is \( \frac{-1}{2x} \).
These separate parts come together to form the full antiderivative of the integrand, which is \[\frac{1}{2}x^3 - x + \frac{-1}{2x},\]plus the constant of integration \(C\). This constant represents the family of all antiderivatives since integration can add arbitrary constants.
Fundamental Theorem of Calculus
The Fundamental Theorem of Calculus links the concepts of differentiation and integration, laying the foundation for evaluating definite integrals. It essentially states that if you have a continuous function over an interval and you know its antiderivative, you can easily find the definite integral.
Here's how it works in simple terms:
  • First, find an antiderivative \( A(x) \) of the integrand.
  • Use this antiderivative to evaluate the definite integral from \(a\) to \(b\) by computing \( A(b) - A(a) \).
For the exercise, the antiderivative \( A(x) \) is \[\frac{1}{2}x^3 - x + \frac{-1}{2x},\]and the definite integral evaluates to \[\int_{1}^{2} \frac{3 x^{4}-2 x^{2}+1}{2 x^{2}} d x = A(2) - A(1).\]
This practical application of the theorem allowed us to find the integral's value efficiently, concluding with \( \frac{7}{4} \). The theorem shows how integration and differentiation are two sides of the same coin.
Integrand Simplification
Before finding an antiderivative, sometimes it's necessary to simplify the integrand for easier integration. This process involves algebraic manipulation of the integrand to form a more straightforward expression.
In the exercise, the integrand \[ \frac{3x^4 - 2x^2 + 1}{2x^2} \] is simplified by dividing each term in the numerator by \(2x^2\):
  • \( \frac{3x^4}{2x^2} = \frac{3}{2}x^2 \)
  • \( \frac{-2x^2}{2x^2} = -1 \)
  • \( \frac{1}{2x^2} = \frac{1}{2}x^{-2} \)
This gives the simplified integrand \[\frac{3}{2}x^2 - 1 + \frac{1}{2x^2},\]ready for integration. Simplification breaks down complex expressions, making the integration process more manageable and reducing the risk of errors when calculating antiderivatives.

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