# More general formula for integrals

• echandler
In summary, the conversation discusses the possibility of a general formula for integrals that accounts for special cases like x^(-1) = ln|x| and u substitutions. The formula int(x^k) = (x^(k+1))/(k+1) for k!=-1 and int(x^(-1))=ln(|x|) is mentioned as a possible combination. Other examples of removable singularities are given, and the idea of defining a function of two variables is proposed. The idea of using limits to define integrals is also discussed, with the general definition being \int_a^b f(x) dx = \lim_{max \Delta x_k \to 0} \sum_{k=1}^n
echandler
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?

You can combine both in a single formula:
"int(x^k) = (x^(k+1))/(k+1) for k!=-1, int(x^(-1))=ln(|x|)"
Apart from that... no.

use limits

$$\int \! x^k \, \mathrm{d}x=\lim_{a \rightarrow k+1} \frac{x^a}{a}+\mathrm{Constant}$$

That is a removable singularity. When we write it in terms of usual functions we appear to be dividing by zero, but we could define a new function without doing so. Other examples include
sin(x)/x
log(1+x)/x
(e^x-1)/x
(sin(tan(x))-tan(sin(x)))/x^7

going the other way we can define the function of two variables
$$\mathrm{f}(x,k)=\int \! x^k \, \mathrm{d}x$$
without any worry about dividing by zero

lurflurf said:
use limits

$$\int \! x^k \, \mathrm{d}x=\lim_{a \rightarrow k+1} \frac{x^a}{a}+\mathrm{Constant}$$
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

$$\int \! x^k \, \mathrm{d}x=\lim_{a \rightarrow k+1} \frac{x^a-1}{a}+\mathrm{Constant}$$

There was a "-1" missing in the numerator, which I added in the quoted equation above. Note that for ##k \neq -1##, the -1/a term can be absorbed into the integration constant.

This is a funny question !
May be, more intuitive if presented on the exponential forme, such as :

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Mute said:
There was a "-1" missing in the numerator, which I added in the quoted equation above.
Ah, that makes sense.

The general definition of the integral that I use is:

$$\int_a^b f(x) dx = \lim_{\text{max} \Delta x_k \to 0} \sum_{k=1}^n f(x_k^*) \Delta x_k$$

Not very useful, but it's definitely general.

## 1. What is a general formula for integrals?

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.

## 2. Why is the general formula for integrals important?

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.

## 3. How do you use the general formula for integrals?

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.

## 4. Are there any special cases for the general formula for integrals?

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.

## 5. What are some applications of the general formula for integrals?

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.

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