Integration of a Square Root including constants

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Char. Limit
Gold Member

Homework Statement

find

$\int3+2\sqrt{1-\frac{x^{2}}{9}} dx$

The Attempt at a Solution

Have tried multiple methods but none seem to work for me!
Have you tried trig substitution? let x = 3sin(s) and then substitute in... you should get an expression that you can integrate, after a few trig tricks of course.

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Char. Limit
Gold Member
It's just an intuitive solution I look at when I see a square root like that. If you substitute it in, you get sqrt(1 - (3 sin2(s))/9), or sqrt(1 - sin2(s)). This, as we know, is equal to cos(s), which is easy to integrate. Then we just need to handle the dx...

But I think once you have x = 3 sin(s), dx should be easy to find. :)

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Char. Limit
Gold Member
so...y=3+2*sqrt(1-(x^2/9))

then... y=y=3+2*sqrt(1-(sin^2(s)/3)) yes?

after that what should occur?
You have a mistake at two places. The sin^2(s) should be (3 sin(s))^2 and the 3 needs to be a 9. Other than that, you're so-far good. After substituting, you should get an easy 3+2cos(s), which of course needs to be multiplied by dx. So while you're doing that, it might be a good idea to get dx in terms of ds.