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nydimaria [60]
3 years ago
5

To break the school’s record, the boys’ time had to be faster than

Mathematics
1 answer:
storchak [24]3 years ago
7 0

Answer:

12 the answer is 12

Step-by-step explanation:

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<img src="https://tex.z-dn.net/?f=%20%5Crm%20%5Cint_%7B0%7D%5E%7B%20%20%5Cpi%20%7D%20%5Ccos%28%20%5Ccot%28x%29%20%20%20%20-%20%2
Nikolay [14]

Replace x with π/2 - x to get the equivalent integral

\displaystyle \int_{-\frac\pi2}^{\frac\pi2} \cos(\cot(x) - \tan(x)) \, dx

but the integrand is even, so this is really just

\displaystyle 2 \int_0^{\frac\pi2} \cos(\cot(x) - \tan(x)) \, dx

Substitute x = 1/2 arccot(u/2), which transforms the integral to

\displaystyle 2 \int_{-\infty}^\infty \frac{\cos(u)}{u^2+4} \, du

There are lots of ways to compute this. What I did was to consider the complex contour integral

\displaystyle \int_\gamma \frac{e^{iz}}{z^2+4} \, dz

where γ is a semicircle in the complex plane with its diameter joining (-R, 0) and (R, 0) on the real axis. A bound for the integral over the arc of the circle is estimated to be

\displaystyle \left|\int_{z=Re^{i0}}^{z=Re^{i\pi}} f(z) \, dz\right| \le \frac{\pi R}{|R^2-4|}

which vanishes as R goes to ∞. Then by the residue theorem, we have in the limit

\displaystyle \int_{-\infty}^\infty \frac{\cos(x)}{x^2+4} \, dx = 2\pi i {} \mathrm{Res}\left(\frac{e^{iz}}{z^2+4},z=2i\right) = \frac\pi{2e^2}

and it follows that

\displaystyle \int_0^\pi \cos(\cot(x)-\tan(x)) \, dx = \boxed{\frac\pi{e^2}}

7 0
2 years ago
Find the 12th term of the sequence -15,-9,-3, 3
Reika [66]
First term (a) = -15
common difference(d)= -9-(-15)
= -9+15=6
So,
12th term(a12)= a +(n-1)d
= -15+(12-1)×6
= -15 + 11×6
= -15+66
= 51
4 0
3 years ago
Find the equations of the lines for the following. Your answer should be in the form y=mx+b.
spin [16.1K]

Answer:

1 ) y = 2x - 1

slope = 2

y intercept is (0,-1)

2) y = -1/2x + 5

slope = -0.5

y intercept  = (0,5)

Step-by-step explanation:

4 0
3 years ago
Solve for X<br> 6x+60x+150=0
hammer [34]

Answer:

<h2><u><em>x = -2.27272727273</em></u></h2><h2><u><em>in fraction</em></u></h2><h2><u><em>x = -25/11</em></u></h2>

<u><em></em></u>

Step-by-step explanation:

Solve for X

6x+60x+150=0

66x = -150

x = -150 : 66

<u><em>x = -2.27272727273</em></u>

<u><em>in fraction</em></u>

<u><em>x = -25/11</em></u>

3 0
1 year ago
Simplify the difference.
sergey [27]
(D) -3w2<span> - 9w - 4
I hope this helped :)
</span>
6 0
3 years ago
Read 2 more answers
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