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Magazine Chapter 5: Problem 8
Evaluate. $$ \int\left(x^{2}-x+2\right) d x $$
Short Answer
Expert verified
\( \frac{x^3}{3} - \frac{x^2}{2} + 2x + C \)
Step by step solution
01
Understand the problem
The task is to find the indefinite integral of the function \(x^2 - x + 2\) with respect to \(x\).
02
Apply the basic integration rules
You need to integrate each term separately. Recall the power rule for integration: \( \int x^n \, dx = \frac{x^{n+1}}{n+1} + C \), where \(C\) is the constant of integration.
03
Integrate \(x^2\)
Using the power rule, \(\int x^2 \, dx\) becomes \( \frac{x^{3}}{3} \).
04
Integrate \(-x\)
Similarly, \(\int (-x) \, dx\) becomes \( \frac{-x^{2}}{2} \).
05
Integrate the constant term
The integral of a constant 2 is \(2x\).
06
Combine all integrals and add the constant of integration
Add all the results together to get: \( \frac{x^3}{3} - \frac{x^2}{2} + 2x + C \).
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Integration Rules
Integration is a fundamental concept in calculus. It helps us find the area under a curve, among other things.
When integrating functions, there are certain rules you need to follow to solve problems effectively.
The main integration rules include:
- Linearity Rule: This states that the integral of a sum of functions is the sum of their integrals. Mathematically, \(\text{if } f(x) = g(x) + h(x), \text{ then } \int f(x) \, dx = \int g(x) \, dx + \int h(x) \, dx\).
- Power Rule: This will be explained in another section, as it鈥檚 a vital rule in integration.
- Constant Rule: If you integrate a constant, say \('a'\), the result will be \(ax + C\), where \('C'\) represents the constant of integration. For example, \(\text{if } a\text{ is 2, then } \int 2 \, dx = 2x + C\).
Power Rule
The Power Rule is incredibly useful for integrating polynomial functions. In simple terms, if you have a term like \(x^n\), the Power Rule tells you how to integrate it.
Here鈥檚 how it works:
Here鈥檚 how it works:
- To integrate \(x^n\), you add 1 to the exponent \(n\).
- You then divide the term by the new exponent \(n+1\).
- For instance, integrating \int x^2 \, dx\ gives you \( \frac{x^{2+1}}{2+1} = \frac{x^3}{3} + C\).
- For \(x^2\): \int x^2 \, dx\ becomes \( \frac{x^3}{3}\).
- For \(-x\): Using the Power Rule gives us \( \frac{-x^2}{2}\).
- And the constant term 2: This is a special case where the Power Rule becomes simply \( 2x\).
Constant of Integration
Whenever you evaluate an indefinite integral, you must include a constant of integration, denoted by \(C\).
But why is this necessary?
But why is this necessary?
- Integration is the reverse process of differentiation. When you differentiate a function, any constant term drops out. Therefore, when integrating, we need to account for any constant that might have existed in the original function.
- This constant \(C\) represents all possible constant values that could complete the integral. It ensures that every antiderivative is accounted for. Thus, \(\int x^2 \, dx\) is not just \(\frac{x^3}{3}\), but \(\frac{x^3}{3} + C\).
