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echandler
- 21
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I was wondering: Is there an even more general formula for the integral than int(x^k) = (x^(k+1))/(k+1) that accounts for special cases like int(x^(-1)) = ln|x| and possibly u substitutions?
For k=-1, that limit is zero for x=0 (which does not fit to the ln), and it is undefined everywhere else. As simple example, consider x=1, where you get the limit of 1/a for a->0.lurflurf said:use limits
$$\int \! x^k \, \mathrm{d}x=\lim_{a \rightarrow k+1} \frac{x^a}{a}+\mathrm{Constant}$$
lurflurf said:use limits
$$\int \! x^k \, \mathrm{d}x=\lim_{a \rightarrow k+1} \frac{x^a-1}{a}+\mathrm{Constant}$$
Ah, that makes sense.Mute said:There was a "-1" missing in the numerator, which I added in the quoted equation above.
The general formula for integrals is ∫f(x) dx = F(x) + C, where f(x) is the integrand, F(x) is the antiderivative of f(x), and C is the constant of integration.
The general formula for integrals is important because it allows us to solve a wide range of mathematical problems involving the calculation of areas and volumes, as well as the evaluation of various physical quantities.
To use the general formula for integrals, you first need to identify the integrand and find its antiderivative. Then, you can plug in the limits of integration and evaluate the integral using the formula ∫f(x) dx = F(x) + C.
Yes, there are special cases for the general formula for integrals, such as when the integrand is a constant or when the limits of integration are infinite. In these cases, the formula simplifies to ∫c dx = cx + C and ∫f(x) dx = ∞, respectively.
The general formula for integrals has many applications in various fields of science and engineering, such as physics, chemistry, and economics. It is used to calculate areas and volumes, as well as to solve differential equations and model real-world phenomena.