Mathematics: Tell me what the final answer to this question would be.

The easiest way to calculate this integral is substitution.Now we can substitute and then:So:...

Tell me what the final answer to this question

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DaFuzzyDude

04.01.2022

First, divide the whole rectangle in fourths and figure out the area of each fourth. Each is 500m².
Second, find the area of the rectangle. 500 x 4 = 2000m² ← that is the area of the rectangle.
Lastly, the perimeter is 25m+25+80+80, because 25x80= 2000.
Answer=210m
:)

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Ответ:

001054213

27.08.2020

Mathematics

The easiest way to calculate this integral is substitution.

$$\int\dfrac{5}{x^\frac{1}{2}+x^\frac{3}{2}}\,dx=5\int\dfrac{1}{x^\frac{1}{2}+(x^\frac{1}{2})^3}\,dx=5\int\dfrac{1}{\sqrt{x}+(\sqrt{x})^3}\,dx=(\star)$

Now we can substitute $u=\sqrt{x}$ and then:

$du=\dfrac{1}{2\sqrt{x}}\,dx\qquad\implies\qquad dx=2\sqrt{x}\,du=2u\,du$

So:

$$(\star)=5\int\dfrac{1}{\sqrt{x}+(\sqrt{x})^3}\,dx=5\int\dfrac{2u}{u+u^3}\,du=10\int\dfrac{1}{1+u^2}\,dx=$

$=10\arctan(u)+C=\boxed{10\arctan(\sqrt{x})+C}$

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