Integration Problem: Solve \int4dy/(1+9y^{2}) 2,0

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In summary, the conversation discusses a problem involving a definite integral with limits. The person has attempted integration by substitution, but has not been successful. They ask for help with determining the derivative of arctan in order to solve the problem. The conversation then continues with suggestions and clarifications, ultimately leading to the correct answer of 4/3 arctan(6).
  • #1
tunabeast
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Homework Statement



Determine [tex]\int[/tex]4dy/(1+9y[tex]^{2}[/tex]) With limits of 2,0.

Homework Equations





The Attempt at a Solution



Have attempted ingtegration by substitution but have had no luck solving this problem. A maths tutor who went over it very quickly established there was a tan in the answer, i have not integrated anything like this before so don't really know where to start.
 
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  • #2
do you know what the derivative of arctan is?
 
  • #3
maybe this looks a little more familiar

[tex]\int_{0}^{2}\frac{4dy}{1+(3y)^{2}}[/tex]
 
Last edited:
  • #4
I have just looked up the definition, can't quite see how it will fit
 
  • #5
[tex]4\int_{0}^{2}\frac{dy}{1+(3y)^{2}}[/tex]

[tex]\mbox{Let u=3y}[/tex]

does it look a little more familiar now?
 
  • #6
I ended up with

4arctan(6)

Am i close?
 
  • #7
no, example

[tex]\frac{d}{dy}\tan^{-1}(3y^{2})=\frac{6ydy}{1+(3y^{2})^{2}}[/tex]
 
  • #8
Hmm i can't seem to get it, when i integrate i get

[tex]\frac{1}{12}tan^{-1}(12)[/tex]
 
  • #9
Ignore that last post, is

[tex]\frac{4}{3}tan^{-1}(6)[/tex] correct?
 
  • #10
do you notice the pattern with my problem?

the angle is [tex]3y^{2}[/tex]

where did my angle and derivative end up when i differentiated?
 
  • #11
tunabeast said:
Ignore that last post, is

[tex]\frac{4}{3}tan^{-1}(6)[/tex] correct?
you're constants are correct but you're angle is wrong. if i took the derivative of your problem it would end up being 0 b/c you're basically saying it's a constant.

[tex]\frac{4}{3}\frac{0}{1+36}[/tex]
 
  • #12
The 6 is just the value of the limits substituted into get a final answer, or is not that what the substituted value would be?
 
  • #13
tunabeast said:
The 6 is just the value of the limits substituted into get a final answer, or is not that what the substituted value would be?
yes that is correct, i did not realize you were already plugging your limits in and evaluating. sorry, miscommunication.
 
  • #14
no problem, thank you very much for your assistance :)
 

1. What is the process for solving an integration problem?

Solving an integration problem involves finding the antiderivative of the given function and evaluating it at the given limits. This can be done using various integration techniques such as substitution, integration by parts, and partial fractions.

2. How do you approach a problem with a fraction in the integrand?

If the integrand contains a fraction, you can use the partial fractions method to simplify it into smaller, more manageable fractions. This will make the integration process easier and more straightforward.

3. Can you explain the purpose of the limits in an integration problem?

The limits in an integration problem define the range over which the function needs to be integrated. They specify the starting and ending points of the integration and help determine the area under the curve of the function.

4. How do you solve a definite integral?

A definite integral is solved by finding the antiderivative of the function and then evaluating it at the given limits. The result is a single numerical value that represents the area under the curve of the function within the specified limits.

5. What are some common mistakes to avoid when solving an integration problem?

Some common mistakes to avoid when solving an integration problem include forgetting to add the constant of integration, using the incorrect limits, and making calculation errors. It is important to double-check your work and be careful with each step of the integration process.

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