Step function integral

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1. Jan 3, 2017

Rectifier

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

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: Jan 3, 2017
2. Jan 3, 2017

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. Jan 3, 2017

Staff: Mentor

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.

4. Jan 3, 2017

Buzz Bloom

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. Jan 3, 2017

Rectifier

Thank you for your help, everyone!