=х²-7х-3х+21-6х²+10х=21-5х²
X-y=1
x+y=-5 /+
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x-y+x+y=1+(-5)
2x=-4
x=-2
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y=-5-x, y=-5-(-2)=-5+2=-3
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/x,y/=/-2,-3/
==============
2cos²x - sin2x + 4cos2x - sin2x = 0
<span>2cos²x - 2sin2x + 4cos2x = 0
</span><span>cos²x - sin2x + 2cos2x = 0
</span>cos²x - 2sinx·cosx + 2(cos²x - sin²x<span>) = 0
</span>3cos²x - 2sinx·cosx - 2sin²x = 0
Однородное уравнение. Делим обе части на cos²x, т.к. cos²x≠0
3 - 2tgx - 2tg²x = 0
2tg²x + 2tgx - 3 = 0
tgx = a
2a² + 2a - 3 = 0
D = 4 + 24 = 28
a = (-2 + 2√7)/4 = (-1 + √7)/2 или a = (-2 - 2√7)/4 = (-1 - √7<span>)/2
tgx = </span> (-1 + √7)/2 tgx = <span> (-1 - √7)/2</span><span>
x = arctg</span> (-1 + √7)/2 + πn x = -arctg (1 + √7) + πk
Найдем абсциссы точек пересечения прямой и параболы. Для этого решим систему уравнений.
![\left \{ {{y=1-2x} \atop {y=x^2-5x-3}} \right. \\1-2x=x^2-5x-3\\x^2-3x-4=0\\(x-4)(x+1)=0\\x_1=4,x_2=-1](https://tex.z-dn.net/?f=+%5Cleft+%5C%7B+%7B%7By%3D1-2x%7D+%5Catop+%7By%3Dx%5E2-5x-3%7D%7D+%5Cright.+%5C%5C1-2x%3Dx%5E2-5x-3%5C%5Cx%5E2-3x-4%3D0%5C%5C%28x-4%29%28x%2B1%29%3D0%5C%5Cx_1%3D4%2Cx_2%3D-1)
Найдем определенный интеграл.
![\int\limits^4_{-1} {(1-2x-(x^2-5x-3))} \, dx =\int\limits^4_{-1} {(1-2x-x^2+5x+3))} \, dx =\\=\int\limits^4_{-1} {(-x^2+3x+4)} \, dx =(- \frac{x^3}{3} + \frac{3x^2}{2}+4x )|^4_{-1}=\\= -\frac{64}{3} + \frac{3*16}{y} +16- \frac{1}{3} - \frac{3}{2} +4=- \frac{65}{3} + \frac{45}{2} +20=\\= \frac{-120+135+120}{6} = \frac{135}{6}](https://tex.z-dn.net/?f=+%5Cint%5Climits%5E4_%7B-1%7D+%7B%281-2x-%28x%5E2-5x-3%29%29%7D+%5C%2C+dx+%3D%5Cint%5Climits%5E4_%7B-1%7D+%7B%281-2x-x%5E2%2B5x%2B3%29%29%7D+%5C%2C+dx+%3D%5C%5C%3D%5Cint%5Climits%5E4_%7B-1%7D+%7B%28-x%5E2%2B3x%2B4%29%7D+%5C%2C+dx+%3D%28-+%5Cfrac%7Bx%5E3%7D%7B3%7D+%2B+%5Cfrac%7B3x%5E2%7D%7B2%7D%2B4x+%29%7C%5E4_%7B-1%7D%3D%5C%5C%3D+-%5Cfrac%7B64%7D%7B3%7D+%2B+%5Cfrac%7B3%2A16%7D%7By%7D+%2B16-+%5Cfrac%7B1%7D%7B3%7D+-+%5Cfrac%7B3%7D%7B2%7D+%2B4%3D-+%5Cfrac%7B65%7D%7B3%7D+%2B+%5Cfrac%7B45%7D%7B2%7D+%2B20%3D%5C%5C%3D+%5Cfrac%7B-120%2B135%2B120%7D%7B6%7D+%3D+%5Cfrac%7B135%7D%7B6%7D+)
Ответ:
![\frac{135}{6}](https://tex.z-dn.net/?f=%5Cfrac%7B135%7D%7B6%7D+)