The antiderivative of the absolute value of x, denoted as ∫|x| dx, is a mathematical concept that relates to finding a function whose derivative is equal to |x|. In other words, we are looking for a function that, when differentiated, results in the absolute value of x.
This particular antiderivative can be somewhat tricky due to the non-differentiability of the absolute value function at x = 0. The absolute value function |x| is defined as x for x ≥ 0 and -x for x < 0. Consequently, the derivative of |x| is positive for x > 0, negative for x < 0, and undefined at x = 0. To find the antiderivative of |x|, we need to divide the integral into two parts, considering the different behavior of the absolute value function for positive and negative values.
Finding the antiderivative of |x| for x > 0
For x > 0, the absolute value function is equal to x. Hence, the antiderivative in this domain becomes:
∫|x| dx = ∫x dx = x^2/2 + C
where C represents the constant of integration.
Finding the antiderivative of |x| for x < 0
For x < 0, the absolute value function transforms into -x. Thus, the antiderivative for this range of x values is: ∫|x| dx = ∫(-x) dx = -x^2/2 + C with C being the constant of integration.
Finding the antiderivative of |x| for x = 0
As previously mentioned, the absolute value function is not differentiable at x = 0. It means we cannot find a continuous antiderivative for |x| that holds true for x = 0. Nonetheless, we can still define a piecewise function that represents an antiderivative of |x|. This function is as follows:
F(x) = (x^2/2 + C1) for x ≥ 0
F(x) = (-x^2/2 + C2) for x < 0
Here, C1 and C2 are constants of integration specific to each interval.
What is the antiderivative of the absolute value of x?
The antiderivative, or integral, of the absolute value of x is given by:
∫|x| dx = (x^2/2 + C) when x ≥ 0
∫|x| dx = (-x^2/2 + C) when x < 0
This piecewise function represents the antiderivative of the absolute value of x.
FAQs:
1. What does an antiderivative represent?
An antiderivative of a function F(x) represents a function whose derivative is equal to the original function f(x).
2. Can the antiderivative of a function have multiple solutions?
No, the antiderivative of a given function has a unique solution up to a constant of integration.
3. If the antiderivative of f(x) is F(x), what is the derivative of F(x)?
If F(x) is the antiderivative of f(x), then the derivative of F(x) is equal to f(x).
4. How is the constant of integration determined?
The constant of integration is determined by evaluating the particular function at a specific point or by using initial conditions, if provided.
5. Is the antiderivative of a function unique?
No, the antiderivative of a function is unique only up to a constant of integration.
6. Can the antiderivative of a continuous function be discontinuous?
No, the antiderivative of a continuous function is always continuous.
7. Are there functions for which an antiderivative cannot be found?
Yes, there are some functions for which an elementary antiderivative cannot be expressed using standard mathematical functions.
8. What other methods can be used to find antiderivatives?
Other methods to find antiderivatives include substitution, integration by parts, and using specialized functions or tables.
9. Can the integral of a function be negative?
Yes, the integral of a function can be negative if the function is negative over the interval of integration.
10. Is the antiderivative of a function always continuous?
No, the antiderivative of a function may not always be continuous, especially if the original function has a discontinuity.
11. Can the antiderivative of a function have a vertical asymptote?
No, the antiderivative of a function does not have vertical asymptotes unless the original function has vertical asymptotes.
12. What is the relationship between the definite integral and antiderivative?
The definite integral of a function represents the accumulation of changes over an interval, whereas the antiderivative represents the original function before differentiation. The fundamental theorem of calculus connects these two concepts by stating that the definite integral of a function can be computed as the difference between the antiderivative evaluated at the endpoints of the interval.