# Integrating 1/-(x-2) and 1/(x-2) - What To Do?

• cmab
In summary, to integrate 1/-(x-2) and 1/(x-2), you can use a substitution method by setting u = x - 2. The only difference between the two expressions is the negative sign, which affects the final result but not the solving process. These expressions cannot be combined, but they have the same domain of all real numbers except x = 2. For special cases where the denominator is raised to a power, a different substitution method and adjustment of the final answer may be necessary.
cmab
Ok, I have to integrate this

[int a=0 b=3] 1 / sqrt[abs(x-2)] dx

what should i do ? do the improper integral of 1/-(x-2) and 1/(x-2) ?

Do you know how to work this out:

$$\int \frac{1}{\sqrt{x-2}}dx$$

If so just split the function you have into the 2 different functions it is composed of and intergrate over the relevant areas.

To integrate 1/-(x-2) and 1/(x-2), you can use the substitution method. Let u = x-2, then du = dx. Substituting this into the integral, we get:

[int a=0 b=3] 1 / sqrt[abs(x-2)] dx = [int a=-2 b=1] 1 / sqrt[abs(u)] du

Now, we can split this into two integrals:

[int a=-2 b=1] 1 / sqrt[abs(u)] du = [int a=-2 b=0] 1 / sqrt[-u] du + [int a=0 b=1] 1 / sqrt du

Using the substitution method again, we can let v = -u for the first integral and v = u for the second integral. This gives us:

[int a=0 b=3] 1 / sqrt[abs(x-2)] dx = [int a=0 b=2] 1 / sqrt[v] dv + [int a=0 b=1] 1 / sqrt du

The first integral can be evaluated as 2*sqrt[v] from 0 to 2, which gives us 2*sqrt[2]. The second integral can be evaluated as 2*sqrt from 0 to 1, which gives us 2*sqrt[1] - 2*sqrt[0] = 2.

Therefore, the final result is 2*sqrt[2] + 2.

## 1. How do I integrate 1/-(x-2) and 1/(x-2)?

To integrate these two expressions, you can use a substitution method. Let u = x - 2. Then, the first expression becomes -1/u and the second expression becomes 1/u. You can then use the rules of integration to solve for u and substitute back in for x.

## 2. What is the difference between 1/-(x-2) and 1/(x-2)?

The only difference between the two expressions is the negative sign in front of 1/u. This negative sign will affect the final result of the integration, but the process for solving remains the same.

## 3. Can I combine 1/-(x-2) and 1/(x-2) into one expression?

No, you cannot combine these two expressions into one. They have different denominators and cannot be added or subtracted together. However, you can integrate them together using the substitution method.

## 4. What is the domain of 1/-(x-2) and 1/(x-2)?

The domain of both expressions is all real numbers except x = 2, as dividing by zero is undefined. In other words, the expressions are defined for all values of x except when x = 2.

## 5. Are there any special cases I should be aware of when integrating 1/-(x-2) and 1/(x-2)?

Yes, you should be aware of the case where the denominator (x-2) is raised to a power. In this case, you will need to use a different substitution method and adjust your final answer accordingly.

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