Need help with finding this antiderivative (Need explanation of steps to find answer)
Hi, I need help figuring out how to work the following problem. The problem has already been worked out, but I don't understand where some of the stuff is coming from.
Here's the problem:
Find the following antiderivative: ∫1/√(-x² - 4x) dx
And here's what has been worked:
Step 1: -x² - 4x = -(x²-4x+4-4)
Step 2: 4-(x+2)²
Step 3: ∫1/√4-(x+2)² dx
Step 4: 1/2∫1/√1-((x+2)/2)² dx
Step 5: (1/2)(2)arcsin((x+2)/2) + C
Okay, so I understand the first step. You just complete the square for the part under the square root. I start to get a little confused in the second step though where it gets simplified. Why is that 4 there in the front? Shouldn't it be a negative 4 instead and be at the end instead of at the front? I understand step 3, just plug in what you simplified down. Step 4 is where I really begin to get confused. Where did all those numbers come from? It looks like everything was divided by something, but I can't figure out by what. It looks like the 4 was divided by 4, the (x+2)² was divided by a 2, and I have no idea where the 1/2 in the front of the problem came from. I don't really understand step 5 either although it really just seems to be plugging some sort of trig identity into the problem.
Re: Need help with finding this antiderivative (Need explanation of steps to find ans
Basically they are factoring out a 4 in the square root which makes it a 2 outside since SQRT(4) = 2 and they just balance it by multiplying by 1/2.