Solving Integration Problems | Get Expert Help Now

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In summary, the conversation discusses an integral and two different methods used to solve it. The error in the second method was forgetting the constant of integration. Both methods ultimately lead to the same answer, but with an added constant.
  • #1
eric_999
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Hey!

I have this integral: ∫((1/2)/(2x-1))dx.

The first time, I did like this: ∫((1/2)/(2x-1))dx = (1/2)∫(1/(2x-1))dx. If I set u = 2x-1, then du = 2dx, so I can rewrite (1/2)∫(1/(2x-1))dx as (1/2)*(1/2)∫(1/u)du = 1/4∫(1/u)du = 1/4ln|u| = 1/4ln|2x-1|.

But when I do like this (I cannot see what I am doing wrong) ∫((1/2)/(2x-1))dx = ∫(1/(4x-2)dx and set u = 4x-2, so du = 4dx, I can rewrite it as 1/4∫(1/u)du = 1/4ln|u| = 1/4ln|4x-2|. What am I doing wrong??

Please help me out! Thanks!
 
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  • #2
The error you made is that you forgot the constant of integration. Include that, and recall that [itex]\ln{ab} = \ln{a} + \ln{b}[/itex] for positive, non-zero a and b, you should be able to see that the methods are equivalent.
 
  • #3
Note that
$$\begin{align}
\frac{1}{4}\ln|4x-2| &= \frac{1}{4}\ln(2|2x-1|) \\
&= \frac{1}{4}(\ln(2) + \ln|2x-1|)\\
&= \frac{1}{4}\ln|2x-1| + \frac{\ln(2)}{4}
\end{align}$$
which differs from your other answer ##\frac{1}{4}|2x-1|## by a constant. Since indefinite integrals can differ by a constant (##+C##), both answers are equally valid.
 
  • #4
Thanks a lot, i understand now!
 

Related to Solving Integration Problems | Get Expert Help Now

What is integration?

Integration is a mathematical process used to find the area under a curve or the accumulation of a quantity over a given interval. It is the reverse of differentiation and is used to solve a wide range of problems in mathematics and science.

What types of integration problems can be solved?

Integration can be used to solve problems related to finding the area under a curve, calculating volumes of irregular shapes, determining the average value of a function, and solving differential equations, among others.

What are the common methods of solving integration problems?

The most common methods include using basic integration formulas, substitution, integration by parts, partial fractions, and trigonometric substitutions. These methods involve breaking down the problem into simpler parts and applying specific techniques to solve them.

Why is it important to solve integration problems accurately?

Solving integration problems accurately is crucial because it allows us to make precise calculations and predictions in various fields such as physics, engineering, economics, and statistics. It also helps us understand the behavior and relationships between different variables in a system.

How can I get help with solving integration problems?

If you are struggling with integration problems, you can seek help from a math tutor, online resources, or a study group. You can also consult with a colleague or professor who has expertise in the subject. Additionally, there are many software programs and calculators that can assist in solving integration problems.

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