1. ## Integration Problem

Last night I did about 30 integration problems that were practice for my upcoming exam. Ironically, the only one I could not get seems to be the easiest one, but I just can't do it. I don't see a method of substitution, and I've tried integration by parts (though it's not my specialty), so I don't know what to do. Any help would be greatly appreciated. Thanks!

It's a definite integral, I don't know how to write it in this forum. It's from 0 to 5:

x/(x+10)

2. Originally Posted by thejabronisayz
Last night I did about 30 integration problems that were practice for my upcoming exam. Ironically, the only one I could not get seems to be the easiest one, but I just can't do it. I don't see a method of substitution, and I've tried integration by parts (though it's not my specialty), so I don't know what to do. Any help would be greatly appreciated. Thanks!

It's a definite integral, I don't know how to write it in this forum. It's from 0 to 5:

x/(x+10)
$\int_0^5 \frac{x}{x + 10}~dx$

The standard way to attack this one is to make the substitution $u = x + 10 \implies du = dx$
$\int_0^5 \frac{x}{x + 10}~dx = \int_{10}^{15} \frac{u - 10}{u}~du = \int_{10}^{15} \left ( 1 - \frac{10}{u} \right )~du$
which you should be able to easily do.

You might find another method to be interesting. I picked it up here on the forum from ThePerfectHacker (I think.)

$\int_0^5 \frac{x}{x + 10}~dx = \int_0^5 \frac{(x + 10) - 10}{x + 10}~dx = \int_0^5 \left ( 1 - \frac{10}{x + 10} \right ) ~ dx$
which you, again, should be able to do easily.

-Dan

3. Based on the way my professor teaches, the second way would probably suffice; we normally haven't done any problems where we change the limits of integration. Thanks!