Askiitians Tutor Team
Last Activity: 5 Months ago
When we talk about even and odd functions, we often refer to their properties in relation to definite integrals, particularly over symmetric intervals like from -a to +a. However, this doesn't mean we can't work with these functions in the context of indefinite integrals. Let's break this down step by step.
Understanding Even and Odd Functions
First, let's clarify what even and odd functions are:
- Even functions: A function f(x) is considered even if f(-x) = f(x) for all x in its domain. Graphically, even functions are symmetric about the y-axis.
- Odd functions: A function f(x) is odd if f(-x) = -f(x) for all x in its domain. These functions have rotational symmetry about the origin.
Definite Integrals and Symmetry
When you evaluate a definite integral of an even function over a symmetric interval, you can simplify the calculation. For example, if you have an even function f(x), then:
∫ from -a to a of f(x) dx = 2 * ∫ from 0 to a of f(x) dx.
For odd functions, the integral over a symmetric interval is zero:
∫ from -a to a of f(x) dx = 0.
Indefinite Integrals of Absolute Functions
Now, let's focus on your question about the indefinite integrals of |x| and |x - 4|. The absolute value function can be tricky because it behaves differently depending on the input value.
Integrating |x|
The function |x| can be expressed piecewise:
- For x ≥ 0, |x| = x
- For x < 0, |x| = -x
To find the indefinite integral of |x|, we can break it into two cases:
∫|x| dx = ∫x dx (for x ≥ 0) = (1/2)x² + C
∫|x| dx = ∫-x dx (for x < 0) = -(1/2)x² + C
Thus, the complete indefinite integral can be expressed as:
∫|x| dx = (1/2)x² + C, for x ≥ 0
∫|x| dx = -(1/2)x² + C, for x < 0
Integrating |x - 4|
Similarly, for |x - 4|, we need to determine where the expression changes:
- For x < 4, |x - 4| = -(x - 4) = 4 - x
- For x ≥ 4, |x - 4| = x - 4
Thus, the indefinite integral can be computed as follows:
For x < 4:
∫|x - 4| dx = ∫(4 - x) dx = 4x - (1/2)x² + C
For x ≥ 4:
∫|x - 4| dx = ∫(x - 4) dx = (1/2)x² - 4x + C
Final Thoughts
In summary, while the properties of even and odd functions are particularly useful for definite integrals over symmetric intervals, they do not limit our ability to integrate functions like |x| and |x - 4| in the indefinite sense. By breaking these functions into their respective cases, we can effectively compute their integrals. This approach not only clarifies the integration process but also enhances our understanding of how absolute values behave across different intervals.
