Efficient Integration of Step Function with Variable Denominator

[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 that's about it. I am not sure how I should continue.

And here is where I get stuck.

 

[Charles Link]

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.

 

[Mark44]

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 that's 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.

 

[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

 

[Rectifier]

Thank you for your help, everyone!