Answer:
Step-by-step explanation:
use the comparison method
3x + 1 = 3x - 3
1 = -3
the expression is false for any value of x and y , so there no solution
I will solve your system by substitution.<span><span>x=<span><span>3y</span>−1</span></span>;<span><span><span>5x</span>−<span>7y</span></span>=19</span></span>Step: Solve<span>x=<span><span>3y</span>−1</span></span>for x:Step: Substitute<span><span>3y</span>−1</span>forxin<span><span><span><span>5x</span>−<span>7y</span></span>=19</span>:</span><span><span><span>5x</span>−<span>7y</span></span>=19</span><span><span><span>5<span>(<span><span>3y</span>−1</span>)</span></span>−<span>7y</span></span>=19</span><span><span><span>8y</span>−5</span>=19</span>(Simplify both sides of the equation)<span><span><span><span>8y</span>−5</span>+5</span>=<span>19+5</span></span><span>(Add 5 to both sides)
</span><span><span>8y</span>=24</span><span><span><span>8y</span>8</span>=<span>248</span></span>(Divide both sides by 8)<span>y=3</span>Step: Substitute3foryin<span><span>x=<span><span>3y</span>−1</span></span>:</span><span>x=<span><span>3y</span>−1</span></span><span>x=<span><span><span>(3)</span><span>(3)</span></span>−1</span></span><span>x=8</span><span>(Simplify both sides of the equation)</span><span>
x=<span><span>8<span> and </span></span>y</span></span>=3
![I=\displaystyle\int\frac x{(1-x^2)^3}\,\mathrm dx](https://tex.z-dn.net/?f=I%3D%5Cdisplaystyle%5Cint%5Cfrac%20x%7B%281-x%5E2%29%5E3%7D%5C%2C%5Cmathrm%20dx)
Haz la sustitución:
![y=1-x^2\implies\mathrm dy=-2x\,\mathrm dx](https://tex.z-dn.net/?f=y%3D1-x%5E2%5Cimplies%5Cmathrm%20dy%3D-2x%5C%2C%5Cmathrm%20dx)
![\implies I=\displaystyle-\frac12\int\frac{\mathrm dy}{y^3}=\frac1{4y^2}+C=\frac1{4(1-x^2)^2}+C](https://tex.z-dn.net/?f=%5Cimplies%20I%3D%5Cdisplaystyle-%5Cfrac12%5Cint%5Cfrac%7B%5Cmathrm%20dy%7D%7By%5E3%7D%3D%5Cfrac1%7B4y%5E2%7D%2BC%3D%5Cfrac1%7B4%281-x%5E2%29%5E2%7D%2BC)
Para confirmar el resultado:
![\dfrac{\mathrm dI}{\mathrm dx}=\dfrac14\left(-\dfrac{2(-2x)}{(1-x^2)^3}\right)=\dfrac x{(1-x^2)^3}](https://tex.z-dn.net/?f=%5Cdfrac%7B%5Cmathrm%20dI%7D%7B%5Cmathrm%20dx%7D%3D%5Cdfrac14%5Cleft%28-%5Cdfrac%7B2%28-2x%29%7D%7B%281-x%5E2%29%5E3%7D%5Cright%29%3D%5Cdfrac%20x%7B%281-x%5E2%29%5E3%7D)
![I=\displaystyle\int\frac{x^2}{(1+x^3)^2}\,\mathrm dx](https://tex.z-dn.net/?f=I%3D%5Cdisplaystyle%5Cint%5Cfrac%7Bx%5E2%7D%7B%281%2Bx%5E3%29%5E2%7D%5C%2C%5Cmathrm%20dx)
Sustituye:
![y=1+x^3\implies\mathrm dy=3x^2\,\mathrm dx](https://tex.z-dn.net/?f=y%3D1%2Bx%5E3%5Cimplies%5Cmathrm%20dy%3D3x%5E2%5C%2C%5Cmathrm%20dx)
![\implies I=\displaystyle\frac13\int\frac{\mathrm dy}{y^2}=-\frac1{3y}+C=-\frac1{3(1+x^3)}+C](https://tex.z-dn.net/?f=%5Cimplies%20I%3D%5Cdisplaystyle%5Cfrac13%5Cint%5Cfrac%7B%5Cmathrm%20dy%7D%7By%5E2%7D%3D-%5Cfrac1%7B3y%7D%2BC%3D-%5Cfrac1%7B3%281%2Bx%5E3%29%7D%2BC)
(Te dejaré confirmar por ti mismo.)
![I=\displaystyle\int\frac x{\sqrt{1-x^2}}\,\mathrm dx](https://tex.z-dn.net/?f=I%3D%5Cdisplaystyle%5Cint%5Cfrac%20x%7B%5Csqrt%7B1-x%5E2%7D%7D%5C%2C%5Cmathrm%20dx)
Sustituye:
![y=1-x^2\implies\mathrm dy=-2x\,\mathrm dx](https://tex.z-dn.net/?f=y%3D1-x%5E2%5Cimplies%5Cmathrm%20dy%3D-2x%5C%2C%5Cmathrm%20dx)
![\implies I=\displaystyle-\frac12\int\frac{\mathrm dy}{\sqrt y}=-\frac12(2\sqrt y)+C=-\sqrt{1-x^2}+C](https://tex.z-dn.net/?f=%5Cimplies%20I%3D%5Cdisplaystyle-%5Cfrac12%5Cint%5Cfrac%7B%5Cmathrm%20dy%7D%7B%5Csqrt%20y%7D%3D-%5Cfrac12%282%5Csqrt%20y%29%2BC%3D-%5Csqrt%7B1-x%5E2%7D%2BC)
![I=\displaystyle\int\left(1+\frac1t\right)^3\frac{\mathrm dt}{t^2}](https://tex.z-dn.net/?f=I%3D%5Cdisplaystyle%5Cint%5Cleft%281%2B%5Cfrac1t%5Cright%29%5E3%5Cfrac%7B%5Cmathrm%20dt%7D%7Bt%5E2%7D)
Sustituye:
![u=1+\dfrac1t\implies\mathrm du=-\dfrac{\mathrm dt}{t^2}](https://tex.z-dn.net/?f=u%3D1%2B%5Cdfrac1t%5Cimplies%5Cmathrm%20du%3D-%5Cdfrac%7B%5Cmathrm%20dt%7D%7Bt%5E2%7D)
![\implies I=-\displaystyle\int u^3\,\mathrm du=-\frac{u^4}4+C=-\frac{\left(1+\frac1t\right)^4}4+C](https://tex.z-dn.net/?f=%5Cimplies%20I%3D-%5Cdisplaystyle%5Cint%20u%5E3%5C%2C%5Cmathrm%20du%3D-%5Cfrac%7Bu%5E4%7D4%2BC%3D-%5Cfrac%7B%5Cleft%281%2B%5Cfrac1t%5Cright%29%5E4%7D4%2BC)
Podemos hacer que esto se vea un poco mejor:
![\left(1+\dfrac1t\right)^4=\left(\dfrac{t+1}t\right)^4=\dfrac{(t+1)^4}{t^4}](https://tex.z-dn.net/?f=%5Cleft%281%2B%5Cdfrac1t%5Cright%29%5E4%3D%5Cleft%28%5Cdfrac%7Bt%2B1%7Dt%5Cright%29%5E4%3D%5Cdfrac%7B%28t%2B1%29%5E4%7D%7Bt%5E4%7D)
![\implies I=-\dfrac{(t+1)^4}{4t^4}+C](https://tex.z-dn.net/?f=%5Cimplies%20I%3D-%5Cdfrac%7B%28t%2B1%29%5E4%7D%7B4t%5E4%7D%2BC)