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nekit [7.7K]
3 years ago
6

How do i find the square root of a number that is not a perfect square?

Mathematics
1 answer:
nlexa [21]3 years ago
7 0
"Calculate the square root of 10 (√ 10) to 4 decimal places.<span>Find the perfect square number closer to 10. 32 = 9 and 42 = 16, so take 3.Divide 10 by 3. 10÷3 = 3.33 (you can round off the answer)Average 3.33 and 3. ( 3.33 + 3)÷2 = 3.1667."
Source-</span>http://burningmath.blogspot.com/2013/12/finding-square-roots-of-numbers-that.html
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Answer:  7\text{Ln}\left(e^{2x}+10\right)+C

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=======================================================

Explanation:

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Apply the derivative and multiply both sides by 7 like so

u = e^{2x}+10\\\\\frac{du}{dx} = 2e^{2x}\\\\7\frac{du}{dx} = 7*2e^{2x}\\\\7\frac{du}{dx} = 14e^{2x}\\\\7du = 14e^{2x}dx\\\\

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This way we can do the following substitutions:

\displaystyle \int \frac{14e^{2x}}{e^{2x}+10}dx\\\\\\\displaystyle \int \frac{1}{e^{2x}+10}14e^{2x}dx\\\\\\\displaystyle \int \frac{1}{u}7du\\\\\\\displaystyle 7\int \frac{1}{u}du\\\\\\

Integrating leads to

\displaystyle 7\int \frac{1}{u}du\\\\\\7\text{Ln}\left(u\right)+C\\\\\\7\text{Ln}\left(e^{2x}+10\right)+C\\\\\\

Be sure to replace 'u' with e^(2x)+10 since it's likely your teacher wants a function in terms of x. Also, do not forget to have the plus C at the end. This is a common mistake many students forget to do.

To verify the answer, you can apply the derivative to it and you should get back to the original integrand of \frac{14e^{2x}}{e^{2x}+10}

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