An integral with a fraction in it-need help finding the antiderivative

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An integral with a fraction in it--need help finding the antiderivative

Homework Statement


I have the integral [tex]\int(\frac{20}{1+x^2} - 2)dx[/tex] where x = ± 3. To solve, I need to find the antiderivative of [tex]\frac{20}{1+x^2} - 2[/tex]

Now, the worksheet actually gives the answer to the problem: The antiderivative is 37.961 or 37.962. However, I'm having trouble actually reaching that answer on my own; I have trouble with the antiderivatives of fractions.


Homework Equations



Lessee... I know that the antiderivative of 1/x is ln x. When it gets any more complex than that I get confused.

The Attempt at a Solution



First, I took [tex]\int(\frac{20}{1+x^2} - 2)dx[/tex] and turned it into [tex]\int(20(1+x^2)^-^1 - 2)dx[/tex]

Then, I tried to take the antiderivative. I thought it would be: [tex]20ln(1+x^2)-2x.[/tex]

Then I plugged in the x values, 3 and -3. With positive 3 plugged in, it came out to 40.052, with a negative 3 it came out to 52.052. I don't even have to finish it off with the subtraction to know that my final answer isn't right and that I missed a step (or two) somewhere. Only question is: What step did I miss? I have a feeling I was supposed to do something with the 20... And I think that there might have been more to taking the antiderivative of (1+x^2)^-1 then just turning it into a natural logarithm. Would someone please help me figure out what I forgot?
 

Answers and Replies

  • #2
Vid
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Try a trig sub or look up derivatives of inverse trig functions.
 
  • #3
Defennder
Homework Helper
2,591
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If you're using a trigo substitution, you may want to use this:
[tex]tan^2\theta + 1 = sec^2\theta[/tex]

That should help you.
 
  • #4
1,341
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Don't be a victim of universal logarithmic differentiation!

[tex]\int\frac{20}{1+x^2}dx[/tex] doesn't equal [tex]20*ln|1+x^2|+c[/tex]
 
Last edited:
  • #5
378
2
If you haven't touched trig int before:

let x = tanu
dx = tanu.secu

substitute these values for x and dx
 
  • #6
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Oh hush! We all know it's 20arctanx - 2x! Cut the kid some slack!

The reason the natural logarithm didn't work was because you need the derivative of the denominator in the numerator. The derivative of 1+x^2 is 2x, and as you can see, it ain't there.
 
  • #7
378
2
Oh hush! We all know it's 20arctanx - 2x! Cut the kid some slack!

The reason the natural logarithm didn't work was because you need the derivative of the denominator in the numerator. The derivative of 1+x^2 is 2x, and as you can see, it ain't there.
I didn't know it three-four years ago :shy:...:cry:
 

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