Solving Integration Problem: I Have No Idea What I'm Doing Wrong

  • Thread starter efekwulsemmay
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In summary, the person is having trouble with the integral and is seeking guidance from a friend. They are using a property of integration that is true for even functions, but is not applicable in this situation.
  • #1
efekwulsemmay
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I have no idea what I am doing wrong. I keep getting one when I should be getting two. It is part of a numerical integration problem. I've got the numerical integration part down which is ironic. The part I am having problems with is finding the actual value of the integral. I need this to find the error of the trapezoid and Simpson's estimations.

Homework Statement


The integral is:
[tex]\int^{\frac{\pi}{2}}_{-\frac{\pi}{2}} \frac{3\cos(t)dt}{\left(2+\sin(t)\right)^{2}} = 2[/tex]

I know it equals 2 cause of the integrate function on my calculator. I am trying to figure out where I am going wrong with my algebra.

Homework Equations


[tex]\int^{a}_{-a} x dx = 2\cdot\int^{a}_{0} x dx[/tex]

[tex]Let u = sin(t) + 2, du = cos(t)dt[/tex]

The Attempt at a Solution


So we start by saying:
[tex]\int^{\frac{\pi}{2}}_{-\frac{\pi}{2}} \frac{3\cos(t)dt}{\left(2+\sin(t)\right)^{2}} [/tex]

We can use the above property of integration to change this to:
[tex]2\cdot\int^{\frac{\pi}{2}}_{0} \frac{3\cos(t)dt}{\left(2+\sin(t)\right)^{2}} [/tex]

We then use u substitution thus we can say:
[tex]x=0 \rightarrow u=2[/tex]

[tex]x= \frac{\pi}{2} \rightarrow u=3[/tex]

so we get:
[tex]2\cdot\int^{3}_{2} \frac{3du}{u^{2}} [/tex]

We can shove the 3 out front and then integrate the resulting [tex]\frac{du}{u^{2}}[/tex]
Thus we get:
[tex]6\cdot\frac{-1}{u}[/tex] Evaluated from 2 to 3.

This goes to:
[tex]6\cdot\left(\frac{-1}{3}-\frac{-1}{2}\right)[/tex]

Which in turn goes to:
[tex]6\cdot\frac{1}{6} = 1[/tex]

I don't know what I am doing wrong. Please help.
 
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  • #2
efekwulsemmay said:

Homework Equations


[tex]\int^{a}_{-a} x dx = 2\cdot\int^{a}_{0} x dx[/tex]

This is true if the function is even. Try not using this when you solve for the integral. ie. use 1 and 3 as your new bounds for your u integral
 
  • #3
jav said:
This is true if the function is even. Try not using this when you solve for the integral. ie. use 1 and 3 as your new bounds for your u integral

Oh bloody hell. :mad: G**d*** m*****f****** piece of s*** integral... grrrr.

Thank you for your help jav. I truly appreciate it. Now I must bang my head against a brick wall somemore :smile:
 

1. What is the purpose of solving integration problems?

Solving integration problems allows us to find the exact value of an integral and understand the relationship between a function and its derivative. It is an important tool in many fields of science and engineering.

2. How do I know if I am doing the integration correctly?

One way to check if you are doing the integration correctly is to take the derivative of your answer and see if it matches the original function. You can also use online integration calculators or ask a peer or instructor for feedback.

3. What are some common mistakes when solving integration problems?

Some common mistakes include forgetting to apply the chain rule, forgetting to add the constant of integration, and making errors in algebraic simplification. It is important to double-check your work and practice regularly to avoid these mistakes.

4. How can I improve my skills in solving integration problems?

Practice is key to improving your skills in solving integration problems. Start with simpler problems and work your way up to more complex ones. Reviewing basic integration techniques and seeking help from tutors or online resources can also be helpful.

5. Can I use integration tables or software to solve integration problems?

Yes, integration tables and software can be useful tools in solving integration problems. However, it is important to understand the concepts and techniques behind integration in order to use these tools effectively and check for accuracy.

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