- #1
Mike s
- 15
- 0
I have already solved it, but I need confirmation:
Are there other ways of proving this?
Thanks in advance!
Are there other ways of proving this?
Thanks in advance!
Mike s said:I have already solved it, but I need confirmation:
Are there other ways of proving this?
Thanks in advance!
Curious3141 said:Your proof is fine (and it's the way I would've done it), except that you should explicitly define your [itex]F(a)[/itex]. You implicitly defined it as an indefinite integral, which means [itex]F(0) = c[/itex], but I would prefer to define [itex]F(a) = \int_0^a f(x) dx[/itex], and include one more intermediary step clarifying that [itex]\int_a^{2a} f(t) dt = \int_0^{2a} f(t) dt - \int_0^a f(t) dt = F(2a) - F(a)[/itex]. This way, I don't have to bother with the [itex]F(0)[/itex] term at all.
An integral is a mathematical concept that represents the area under a curve in a graph. It is typically denoted by ∫ and is used to find the total value of a quantity over a certain interval.
In order to prove a property of an integral, one must use mathematical techniques such as substitution, integration by parts, or the fundamental theorem of calculus. These techniques allow you to manipulate the integral in a way that leads to the desired property.
One common property of integrals is the linearity property, which states that the integral of a sum is equal to the sum of the integrals. For example, if we have the integral of (x^2 + 2x)dx, we can use the linearity property to split it into the integral of x^2dx + the integral of 2xdx, making it easier to solve.
A definite integral has specific limits of integration, meaning it is evaluated over a specific interval. An indefinite integral does not have limits and represents a family of functions that differ by a constant. In other words, a definite integral gives a specific value while an indefinite integral gives a function.
Yes, there are several other properties of integrals, including the product rule, quotient rule, and change of variables. These properties allow you to manipulate integrals in different ways to solve them more easily or to express them in terms of other variables.