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The limit from 1 to 2 of the given antiderivative is; -0.19865
<h3>What is the Limit of the Integral?</h3>
We are given the antiderivative of f(x) as sin(1/(x² + 1)). Thus, to find the limit from 1 to 2, we will solve as;
![\int\limits^2_1 {(sin\frac{1}{x^{2} + 1} )} \, dx = {(sin\frac{1}{2^{2} + 1} )} - {(sin\frac{1}{1^{2} + 1} )}](https://tex.z-dn.net/?f=%5Cint%5Climits%5E2_1%20%7B%28sin%5Cfrac%7B1%7D%7Bx%5E%7B2%7D%20%2B%201%7D%20%29%7D%20%5C%2C%20dx%20%3D%20%7B%28sin%5Cfrac%7B1%7D%7B2%5E%7B2%7D%20%2B%201%7D%20%29%7D%20%20-%20%7B%28sin%5Cfrac%7B1%7D%7B1%5E%7B2%7D%20%2B%201%7D%20%29%7D)
⇒ (sin ¹/₅) - (sin ¹/₂)
⇒ 0.19866 - 0.47942
⇒ -0.19865
Complete Question is;
If sin(1/(x² + 1)) is an anti derivative for f(x), then what is the limit of f(x)dx from 1 to 2?
Read more about integral limits at; brainly.com/question/10268976
Based on the calculations, the radius of this circle with the given equation is equal to 5 units.
<u>Given the following data:</u>
<h3>The equation of a circle.</h3>
Mathematically, the standard form of the equation of a circle is given by;
![(x-h)^2+(y-k)^2=r^2](https://tex.z-dn.net/?f=%28x-h%29%5E2%2B%28y-k%29%5E2%3Dr%5E2)
<u>Where:</u>
- h and k represents the coordinates at the center.
- r is the radius of a circle.
Expanding the equation, we have:
![x^2+y^2-2hx-2ky+(h^2+k^2-r^2)=0\\\\r=\sqrt{h^2+k^2}](https://tex.z-dn.net/?f=x%5E2%2By%5E2-2hx-2ky%2B%28h%5E2%2Bk%5E2-r%5E2%29%3D0%5C%5C%5C%5Cr%3D%5Csqrt%7Bh%5E2%2Bk%5E2%7D)
Comparing the terms, we have:
![-2hx = 8x\\\\-2h=8\\\\h=\frac{8}{-2}](https://tex.z-dn.net/?f=-2hx%20%3D%208x%5C%5C%5C%5C-2h%3D8%5C%5C%5C%5Ch%3D%5Cfrac%7B8%7D%7B-2%7D)
h = -4
![-2ky = -6y\\\\-2k=6\\\\k=\frac{6}{-2}](https://tex.z-dn.net/?f=-2ky%20%3D%20-6y%5C%5C%5C%5C-2k%3D6%5C%5C%5C%5Ck%3D%5Cfrac%7B6%7D%7B-2%7D)
k = 3.
Now, we can determine the radius:
![r=\sqrt{-4^2+3^2}\\\\r=\sqrt{16+9} \\\\r=\sqrt{25}](https://tex.z-dn.net/?f=r%3D%5Csqrt%7B-4%5E2%2B3%5E2%7D%5C%5C%5C%5Cr%3D%5Csqrt%7B16%2B9%7D%20%5C%5C%5C%5Cr%3D%5Csqrt%7B25%7D)
r = 5 units.
Read more on circle here: brainly.com/question/14078280