applying linearity?

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lethalasian

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Feb 14, 2020
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Stupid question but. I am doing arc length formula but I don't know how they simplified.
How do you simplify Integral of sqrt [[e^(2x) -2 +e^(-2x)]/4) +1]
the solution is 1/2 integral sqrt(e^2x+2+e^-2x)
Why did it go from -2 to + 2 inside the sqrt after they took the 1/2 out?
and for the +1 you can just take outside because sqrt 1 = 1?
 
Inside the square root you have \(\displaystyle \frac{e^{2x}- 2+ e^{-2x}}{4}+ 1= \frac{e^{2x}}{4}- \frac{1}{2}+ \frac{e^{-2x}}{4}+ 1= \frac{e^{2x}}{4}+ \frac{1}{2}+ \frac{e^{-2x}}{4}\).

Factoring out "1/4" that is \(\displaystyle \frac{1}{4}(e^{2x}+ 2+ e^{-2x})\) and, of course, taking "1/4" out of the square root gives "1/2".
 
\(\displaystyle \dfrac{-2}{4} + 1 = \dfrac{-2}{4} + \dfrac{4}{4} = \dfrac{-2+4}{4} = \dfrac{2}{4} = \dfrac{1}{4}*2\)

If you are having trouble with this arithmetic problem you will have a hard time in calculus
 
@Jomo, I suspect that the problem was not failure to add fractions correctly, but failure to look at details. The OP was trying to understand a provided solution, and probably just had a wrong expectation. Math requires slow, active reading, rather than skimming as we can do elsewhere.

I find that many students get into trouble trying to "reverse engineer" a tersely presented solution, because they expect each step shown to be immediately understandable. What they need to learn is to do the work themselves for each step, and then compare their result with the next step shown. In this case, that reveals that much more happened than just a -2 becoming a +2.

@lethalasian, I hope that each time you found that you have "missed a small thing", you will take the time to discover what sort of thinking it would have taken to see that small thing. Even if my suggestion here is not exactly what happened, you should be able to improve your approach by learning from mistakes.
 
I think the problem is staring at the "smart phone" and "avoiding" use of pencil and paper.
 
In a calc mindset, I tried simplifying without doing the addition. I took calc 1 5 years ago and jumped into calc 2 recently so trying to cram and catch up. Off course using paper and pencil just comparing my steps to the answer but doing some many problems making my brain fried
 
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