Problem 12 - Basic Engineering Mathematics

Question

Evaluate: (a) \(\int_{0}^{2} x(3+2 x) \mathrm{d} x\) (b) \(\int_{-1}^{1}\left(\frac{x^{4}-5 x^{2}+x}{x}\right) \mathrm{d} x\)

Step-by-Step Solution

Answer

(a) \(\frac{34}{3}\), (b) \(\frac{5}{2}\).

Step 1: Expand the integrand for Part (a)

First, expand the expression inside the integral for Part (a): \[ x(3 + 2x) = 3x + 2x^2. \]Now, the integral becomes:\[ \int_{0}^{2} (3x + 2x^2) \, \mathrm{d}x. \]

Step 2: Integrate the expression for Part (a)

Integrate each term separately:- For \(3x\), the integral is \(\frac{3x^2}{2}\).- For \(2x^2\), the integral is \(\frac{2x^3}{3}\).So, the integral of the expanded expression is:\[ \left[\frac{3x^2}{2} + \frac{2x^3}{3}\right]_{0}^{2}. \]

Step 3: Evaluate the definite integral for Part (a)

Substitute the upper limit \(x = 2\) and the lower limit \(x = 0\) into the integral:1. At \(x = 2\), the expression evaluates to: \[ \frac{3(2)^2}{2} + \frac{2(2)^3}{3} = 6 + \frac{16}{3}. \]2. At \(x = 0\), the expression evaluates to \(0\).The value of the integral is:\[ 6 + \frac{16}{3} = \frac{34}{3}. \]

Step 4: Simplify the integrand for Part (b)

Simplify the integrand for Part (b) by dividing each term by \(x\):\[ \frac{x^4}{x} - \frac{5x^2}{x} + \frac{x}{x} = x^3 - 5x + 1. \]So, the integral becomes:\[ \int_{-1}^{1} (x^3 - 5x + 1) \, \mathrm{d}x. \]

Step 5: Integrate the expression for Part (b)

Integrate each term separately:- For \(x^3\), the integral is \(\frac{x^4}{4}\).- For \(-5x\), the integral is \(-\frac{5x^2}{2}\).- For \(1\), the integral is \(x\).So, the integral is:\[ \left. \left(\frac{x^4}{4} - \frac{5x^2}{2} + x\right) \right|_{-1}^{1}. \]

Step 6: Evaluate the definite integral for Part (b)

Substitute the upper limit \(x = 1\) and the lower limit \(x = -1\) into the integral:1. At \(x = 1\), the expression evaluates to: \[ \frac{1^4}{4} - \frac{5(1)^2}{2} + 1 = \frac{1}{4} - \frac{5}{2} + 1 = \frac{-3}{4}. \]2. At \(x = -1\), the expression evaluates to: \[ \frac{(-1)^4}{4} - \frac{5(-1)^2}{2} + (-1) = \frac{1}{4} - \frac{5}{2} - 1 = \frac{-13}{4}. \]The value of the integral is:\[ \frac{-3}{4} - \left(\frac{-13}{4}\right) = \frac{10}{4} = \frac{5}{2}. \]

Key Concepts

Integration Techniques

Integration is an essential concept in calculus, often used to find areas under curves and solve problems involving rates of change. Various techniques exist to solve integrals more effectively. Here's a look at some basic methods:

• Integration by Substitution: This technique simplifies an integral by changing variables, making the integral resemble a more familiar form.
• Integration by Parts: This technique is useful when the integrand is a product of two functions. It is derived from the product rule of differentiation.
• Partial Fraction Decomposition: Used when dealing with rational functions, where the aim is to break the function into simpler fractions that are easier to integrate.

In the given problem, simple polynomial functions were involved. Therefore, basic antiderivatives were directly applied. As you master various integration techniques, you'll become more equipped to tackle complex integrals.

Evaluating Integrals

Evaluating definite integrals involves two main steps: finding the antiderivative and applying the limits of integration. Let’s walk through these steps using the example provided.For Part (a), after simplifying the integrand and finding derivatives, the antiderivative was determined as \[ \frac{3x^2}{2} + \frac{2x^3}{3} \]. Once you have the antiderivative, you calculate its difference at the upper and lower limits of the integral:

• First, substitute the upper limit (2), resulting in \[ 6 + \frac{16}{3} \].
• Then, substitute the lower limit (0), which results in \[ 0 \].
• Subtract the value at the lower limit from the upper limit to find the integral's value \[ \frac{34}{3} \].

This straightforward approach can handle simple polynomials effectively. It emphasizes understanding the function's behavior over a specified interval, represented through graphical or numeric interpretations.

Expanding Expressions

Expanding expressions involves breaking down an expression into simpler components or terms that are easier to manage, especially when integrating. This method allows for direct application of basic integration rules.For example, in Part (a), the original expression inside the integrand was \( x(3+2x) \). To expand:

• Multiply \( x \) with each term inside the parentheses:
• \( x \times 3 = 3x \) and \( x \times 2x = 2x^2 \).
• The expression then becomes \( 3x + 2x^2 \).

Expanding expressions simplifies the integration process. It allows each term in the polynomial to be integrated separately, streamlining the process. This skill is crucial as it turns complex, condensed expressions into manageable, individual polynomials.

Simplifying Expressions

Simplifying expressions is about reducing them to their simplest form. This step often involves canceling terms and combining like terms, which makes evaluating integrals more manageable.In Part (b), the original expression inside the integrand \( \frac{x^4 - 5x^2 + x}{x} \) was simplified by dividing each term in the numerator by \( x \):

• \( \frac{x^4}{x} = x^3 \)
• \( \frac{-5x^2}{x} = -5x \)
• \( \frac{x}{x} = 1 \)

Putting these together, the simplified expression was \( x^3 - 5x + 1 \). By simplifying the integrand, you not only make the integral easier to evaluate but also reduce the potential for error as each term is dealt with independently. Simplification is often the first step towards making a complex problem more predictable and solvable.