Integrating Fractions with Substitution

In summary, the conversation discusses a problem involving integration and the use of substitution to solve it. The person initially solves the problem using substitution and obtains an answer that is slightly different from the one given by Wolfram Alpha. However, it is explained that the two answers are essentially the same and can be checked by taking the derivative or differentiating by hand.
  • #1
cp255
54
0
So the problem is ∫(6x+5)/(2x+1)dx. I know the proper way to solve this is to long divide these two expressions and then solve. However, I tried doing it with substitution.

u = 2x+1
dx = du/2
I then reasoned that 3u + 2 = 6x+5 since 3(2x+1) + 2 = 6x+3+2 = 6x+5 so I substituted it on top.
(1/2)∫(3u+2)/u du
Solving this integral gives me (3/2)u + ln|u| + C which equals (6x+3)/2 + ln|2x+1| + C.

However, Wolfram Alpha says the answer is 3x + ln|2x+1| + C. I don't understand what I did wrong or why I can't do what I did.
 
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  • #2
It's the same answer. The 3/2 term gets absorbed into the + C term because 3/2 is just a constant. You did the problem correctly.
 
  • #3
Well that makes sence. It is sometimes hard to check my calculus work especially with computers since there are so many different forms an answer can take.
 
  • #4
A useful trick I do for integration problems is plug the answer in and take the derivative. If you get the original problem back, you have the correct answer. Wolfram doesn't always compute it the same way as you and might simplify it differently. This is especially apparent if you do a trig substitution and have a theta hanging around in your answer. Using the triangle you made for the trig substitution, you can have 6 different answers for the value of theta that are the same (arcsin, arccos, etc).

You can also differentiate by hand if the answer isn't too complicated to check yourself.
 

What is substitution in fractions?

Substitution in fractions is a method of solving equations that involve fractions by replacing a variable with an equivalent expression. This can simplify the equation and make it easier to solve.

How do you substitute fractions in an equation?

To substitute fractions in an equation, you must first identify the variable and its equivalent expression. Then, you replace the variable with the equivalent expression in the equation. This can be done for both the numerator and denominator of a fraction.

Can substitution be used for all types of fractions?

Yes, substitution can be used for all types of fractions, including proper fractions, improper fractions, and mixed numbers. The key is to identify the variable and its equivalent expression correctly.

What are the benefits of using substitution in fractions?

Using substitution in fractions can make solving equations easier and more efficient. It can also help to simplify complex fractions and make them more manageable.

Are there any limitations to using substitution in fractions?

While substitution can be a powerful tool in solving equations with fractions, it may not always be the most efficient method. In some cases, it may be more practical to use other methods, such as finding common denominators or cross-multiplying.

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