- #1

adelin

- 32

- 0

What rule have they used to change the integral from 2x/(x+1)^2?

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In summary, the integral was changed by using the technique of adding zero, where the expression was rewritten in a way that allowed for a cancellation of terms. This is a common trick used in integration, along with other techniques such as u-substitution.

- #1

adelin

- 32

- 0

What rule have they used to change the integral from 2x/(x+1)^2?

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- #2

eumyang

Homework Helper

- 1,347

- 11

[itex]\int \frac{3x}{(x-2)^2} dx[/itex]

I would subtract and add 6:

[itex]= \int \frac{3x - 6 + 6}{(x-2)^2} dx[/itex]

[itex]= \int \frac{3x - 6}{(x-2)^2} dx + \int \frac{6}{(x-2)^2} dx[/itex]

[itex]= \int \frac{3(x-2)}{(x-2)^2} dx + \int \frac{6}{(x-2)^2} dx[/itex]

... etc.

- #3

- 41,907

- 10,110

I assume you mean 2x/(x-1)^2.adelin said:What rule have they used to change the integral from 2x/(x+1)^2?

Are you asking how it's valid (isn't it obviously valid?) or how they thought to do that?

- #4

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In particular, they added 0 (which doesn't change anything) to the numerator, but with "zero" written as "2-2". There are lots of very creative ways to add zero.eumyang said:It's called "adding zero."

A related technique (not used here) is to multiply by one, where once again you can be very creative in the way in which you write "one".

- #5

Mark44

Mentor

- 37,743

- 10,086

With an expression you are much more limited. One thing you can do is add 0 (in some form) or multiply by 1 (also in some form). You can also factor the expression if that seems useful to do, or expand it, if the situation calls for that operation.

- #6

- 9,568

- 775

u = x-1,~x = u+1,~du=dx$$ in the first place giving$$

\int \frac {2(u+1)}{u^2}~du = \int\frac 2 u + \frac 2 {u^2}~du$$leading to the same answer.

An integral is a mathematical concept used to calculate the area under a curve or the accumulation of a quantity over a given interval. It is important because it allows us to solve a wide range of problems in various fields such as physics, engineering, and economics.

Integrals can be changed or manipulated using different rules such as the power rule, product rule, quotient rule, and chain rule. These rules help us simplify the integral and make it easier to solve.

We change integrals to make them easier to solve or to find an equivalent expression that is more useful for solving a particular problem. Sometimes, changing an integral can also help us to better understand the underlying concepts and relationships.

Some commonly used rules for changing integrals include the power rule, substitution rule, integration by parts, trigonometric substitutions, and partial fraction decomposition. These rules are based on algebraic and trigonometric identities that help us simplify the integrand.

Yes, there can be restrictions or limitations when changing integrals. For example, some integrals may not have an elementary solution and may require advanced techniques such as numerical methods or approximation. Also, when using substitution, we must ensure that the new variable is a valid substitution for the original variable.

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