Solving the Integral ∫dx/(1-x)

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You are solving the same integral using two different methods, and getting different answers. This is due to a property of the absolute value, |a - b| = |b - a|. Both methods are correct.
  • #1
Fernando Rios
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Homework Statement
Solve the following integral
Relevant Equations
∫dx/(1-x)
I solved the integral by two different methods and I get different answers.

Method 1:
∫dx/(1-x) = -∫-dx/(1-x), u=1-x, du=-dx

∫dx/(1-x) = -∫du/u = -ln|u| = -ln|1-x|

Method 2:
∫-dx/(x-1) = -∫dx/(x-1), u=x-1, du=dx

∫-dx/(x-1) = -∫du/u = -ln|u| = -ln|x-1|

What am I doing wrong?
 
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  • #2
Try calculating both quantities for, say, ##x=3##.
 
  • #3
Fernando Rios said:
Homework Statement:: Solve the following integral
Relevant Equations:: ∫dx/(1-x)

I solved the integral by two different methods and I get different answers.

Method 1:
∫dx/(1-x) = -∫-dx/(1-x), u=1-x, du=-dx

∫dx/(1-x) = -∫du/u = -ln|u| = -ln|1-x|

Method 2:
∫-dx/(x-1) = -∫dx/(x-1), u=x-1, du=dx

∫-dx/(x-1) = -∫du/u = -ln|u| = -ln|x-1|

What am I doing wrong?
At heart, your question really isn't about calculus -- it's about a property of the absolute value.
|a - b| = |b - a|, right?
 
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  • #4
Fernando Rios said:
What am I doing wrong?
Nothing
 
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FAQ: Solving the Integral ∫dx/(1-x)

1. What is the process for solving the integral ∫dx/(1-x)?

The process for solving this integral involves using the substitution method. Let u = 1-x, then du = -dx. The integral then becomes ∫-du/u, which can be solved using the natural logarithm function. The final answer is -ln|1-x| + C.

2. Why is the substitution method used for solving this integral?

The substitution method is used because it allows us to simplify the integral into a more manageable form. In this case, substituting u for 1-x helps us to eliminate the denominator and make the integration process easier.

3. Can this integral be solved using other methods?

Yes, this integral can also be solved using partial fractions or integration by parts. However, the substitution method is the most straightforward and efficient approach for this particular integral.

4. What are the limits of integration for this integral?

Since the integral does not specify any limits, the limits of integration can be assumed to be from -∞ to +∞. However, if the integral is part of a larger problem or equation, the limits may be specified based on the context.

5. Are there any special cases for solving this integral?

Yes, if the integral is ∫dx/(1-x) with limits from 0 to 1, the answer would be -ln|1-x| evaluated at 0 and 1, which simplifies to -ln(0) + ln(1) = ∞. This is known as a divergent integral and indicates that the area under the curve is infinite.

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