Calculus Problem: Find Integral of 1/(1-x)^2 dx

In summary, the conversation is about someone asking for help with a calculus problem involving an indefinite integral. The integral in question is 1/(1-x)^2 dx = x/(1-x), but the experts in the conversation point out that this is not correct. They suggest differentiating to prove the integral is correct and remind the person to include a constant term in their answer. The person asks for clarification on how to solve the problem and mentions that their book specifically asks them to prove the integral is correct.
  • #1
gigi9
40
0
Calculus help please!

Plz help me do the problem below. thanks a lot.
Show that the following integral is CORRECT:
Indefinite Integral of 1/(1-x)^2 dx = x/(1-x)
 
Physics news on Phys.org
  • #2
I'm afraid no one will be able to "Show that the following integral is CORRECT: Indefinite Integral of 1/(1-x)^2 dx = x/(1-x)"

because it isn't.

Did you even try differentiating to see if it was correct?
 
  • #3
As HallsofIvy stated in his post, and I stated when you asked the similar question at https://www.physicsforums.com/showthread.php?threadid=6977 , the simplest way to prove an indefinte integral correct is to differentiate.


Incidentally, whenever you're doing an indefinite integral, you have to include a constant term; so if this integral is correct, you should write

∫ 1/(1-x)^2 dx = x/(1-x) + C

(for the record, once you add the "+ C", the answer is correct)


Incidentally, gigi9, how did you come to ask us this question? Does your book say "prove this integral is correct", or did it ask you to find the integral and you derived the RHS on your own?
 
  • #4
My book say "SHOW THAT THE FOLLOWING INTEGRAL IS CORRECT"
 

1. What is the purpose of finding the integral of 1/(1-x)^2 dx?

The integral of 1/(1-x)^2 dx is used to find the area under the curve of the function 1/(1-x)^2. This can be useful in various applications, such as determining the total displacement of an object given its velocity over time.

2. How do you solve this integral?

To solve this integral, you can use the substitution method. Let u = 1-x, then dx = -du. The integral becomes ∫ 1/u^2 (-du) which simplifies to -∫ 1/u^2 du. This can then be solved using the power rule for integrals, resulting in -1/u + C. Finally, substitute back u = 1-x to get the final answer of -1/(1-x) + C.

3. Can this integral be solved using any other methods?

Yes, this integral can also be solved using the partial fractions method. The function can be rewritten as 1/(1-x)^2 = A/(1-x) + B/(1-x)^2, where A and B are constants. By equating the coefficients of similar terms on both sides, you can solve for A and B and then integrate each term separately.

4. Is there any significance to the constant C in the final answer?

Yes, the constant C is known as the constant of integration and it accounts for all possible solutions to the integral. This is because when you take the derivative of -1/(1-x) + C, you get 1/(1-x)^2, but you can add any constant value to this derivative and it will still be equivalent. Therefore, the constant C is necessary to include in the final answer.

5. Can this integral be used to solve real-world problems?

Yes, this integral can be used to solve real-world problems in various fields such as physics, engineering, and economics. For example, in physics, it can be used to find the total distance traveled by an object with a given velocity function. In economics, it can be used to calculate the total revenue generated by a company with a given demand function. Overall, the integral of 1/(1-x)^2 dx has many practical applications in different fields.

Similar threads

Replies
31
Views
753
Replies
12
Views
1K
  • Calculus
Replies
3
Views
2K
  • Calculus
Replies
6
Views
1K
Replies
20
Views
2K
Replies
8
Views
1K
Replies
14
Views
615
Replies
1
Views
814
Replies
4
Views
1K
Back
Top