Step function integral

  • #1
Rectifier
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The problem
I want to calculate ## \int^6_{-6} \frac{g(x)}{2+g(x)} \ dx ## for the step function below.


2YSK7nM.jpg



The attempt
I started with rewriting the function as with the help of long-division
## \int^6_{-6} \frac{g(x)}{2+g(x)} \ dx = \int^6_{-6} 1 \ dx - 2\int^6_{-6} \frac{1}{g(x)+2} \ dx##

I know that ##\int^6_{-6} 1 \ dx = 12## but thats about it. I am not sure how I should continue.

And here is where I get stuck.
 
Last edited:

Answers and Replies

  • #2
Charles Link
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If you look closely at ## g(x) ##, it takes on constant values for various intervals. You need to break up the integral from -6 to 6 into these various segments.
 
  • #3
34,948
6,705
The problem
I want to calculate ## \int^6_{-6} \frac{g(x)}{2+g(x)} \ dx ## for the step function below.


2YSK7nM.jpg



The attempt
I started with rewriting the function as with the help of long-division
## \int^6_{-6} \frac{g(x)}{2+g(x)} \ dx = \int^6_{-6} 1 \ dx - 2\int^6_{-6} \frac{1}{g(x)+2} \ dx##

I know that ##\int^6_{-6} 1 \ dx = 12## but thats about it. I am not sure how I should continue.

And here is where I get stuck.
This shouldn't be too difficult. On the interval [-6, -4], g(x) = -1, so g(x) + 2 = 1. What is ##\int_{-6}^{-4} \frac 1 1 dx##? Do the same for the other intervals.
 
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  • #4
Buzz Bloom
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Hi Rectifier:

I suggest breaking the integral into six pieces, one piece for each step. For each piece, g(x) has a specific constant value, so the integrand is a specific constant.

Hope this helps.

Regards,
Buzz
 
  • #5
Rectifier
Gold Member
313
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Thank you for your help, everyone!
 
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