Does the integral of 2/(x-6)^2 from 0 to 8 converge or diverge?

[graycolor]

integral 2/(x-6)^2 respects to x from 0 to 8. Shouldn't the answer to this be converges and is =-4/3. The true answer to this is it diverges towards infinity... can someone please explain.

 

[statdad]

Two things to consider:

1. isn't there a number between $$0$$ and $$8$$ that could cause a problem?

2. The integrand $$2/{(x-6)^2}$$ is positive throughout the interval of integration,
so how could the integral be negative?

 

[graycolor]

I think I've found the problem there should be an asymptote when x=6. My teacher never taught me to look for these.

 

[statdad]

Whether you are graphing, integrating, differentiating, or simply contemplating the function and its domain for their inherent beauty you should always look for the possible existence, and influence of, an asymptote for rational functions.