2x-y=11 ,6x-3y=15
4x+4y=26
-4y. -4y
4x=22
4. 4. divided
x=5.5 if its wrong and fix something that's wrong
Answer:
![\int \frac{3e^x}{e^{2x}+2e^x+1}dx=-\frac{3}{e^x+1}+C](https://tex.z-dn.net/?f=%5Cint%20%5Cfrac%7B3e%5Ex%7D%7Be%5E%7B2x%7D%2B2e%5Ex%2B1%7Ddx%3D-%5Cfrac%7B3%7D%7Be%5Ex%2B1%7D%2BC)
Step-by-step explanation:
To find this integral
you must:
1. Take the constant out:
![3\cdot \int \frac{e^x}{e^{2x}+2e^x+1}dx](https://tex.z-dn.net/?f=3%5Ccdot%20%5Cint%20%5Cfrac%7Be%5Ex%7D%7Be%5E%7B2x%7D%2B2e%5Ex%2B1%7Ddx)
2. Factor ![{e^{2x}+2e^x+1}=(e^{x}+1)^2](https://tex.z-dn.net/?f=%7Be%5E%7B2x%7D%2B2e%5Ex%2B1%7D%3D%28e%5E%7Bx%7D%2B1%29%5E2)
![3\cdot \int \frac{e^x}{\left(e^x+1\right)^2}dx](https://tex.z-dn.net/?f=3%5Ccdot%20%5Cint%20%5Cfrac%7Be%5Ex%7D%7B%5Cleft%28e%5Ex%2B1%5Cright%29%5E2%7Ddx)
3. Apply u-substitution ![u=e^x+1](https://tex.z-dn.net/?f=u%3De%5Ex%2B1)
![3\cdot \int \frac{e^x}{\left(u\right)^2}dx](https://tex.z-dn.net/?f=3%5Ccdot%20%5Cint%20%5Cfrac%7Be%5Ex%7D%7B%5Cleft%28u%5Cright%29%5E2%7Ddx)
![u=e^x+1\\du=e^xdx\\dx=\frac{du}{e^x}](https://tex.z-dn.net/?f=u%3De%5Ex%2B1%5C%5Cdu%3De%5Exdx%5C%5Cdx%3D%5Cfrac%7Bdu%7D%7Be%5Ex%7D)
![3\cdot \int \frac{e^x}{\left(u\right)^2}\frac{du}{e^x} \\\\3\cdot \int \frac{1}{u^2}du](https://tex.z-dn.net/?f=3%5Ccdot%20%5Cint%20%5Cfrac%7Be%5Ex%7D%7B%5Cleft%28u%5Cright%29%5E2%7D%5Cfrac%7Bdu%7D%7Be%5Ex%7D%20%5C%5C%5C%5C3%5Ccdot%20%5Cint%20%5Cfrac%7B1%7D%7Bu%5E2%7Ddu)
![3\cdot \int \:u^{-2}du](https://tex.z-dn.net/?f=3%5Ccdot%20%5Cint%20%5C%3Au%5E%7B-2%7Ddu)
4. Apply the Power Rule ![\int x^adx=\frac{x^{a+1}}{a+1},\:\quad \:a\ne -1](https://tex.z-dn.net/?f=%5Cint%20x%5Eadx%3D%5Cfrac%7Bx%5E%7Ba%2B1%7D%7D%7Ba%2B1%7D%2C%5C%3A%5Cquad%20%5C%3Aa%5Cne%20-1)
![3\cdot \frac{u^{-2+1}}{-2+1}](https://tex.z-dn.net/?f=3%5Ccdot%20%5Cfrac%7Bu%5E%7B-2%2B1%7D%7D%7B-2%2B1%7D)
5. Substitute back ![u=e^x+1](https://tex.z-dn.net/?f=u%3De%5Ex%2B1)
![3\cdot \frac{\left(e^x+1\right)^{-2+1}}{-2+1}=3\cdot -\left(e^x+1\right)^{-1}=-\frac{3}{e^x+1}](https://tex.z-dn.net/?f=3%5Ccdot%20%5Cfrac%7B%5Cleft%28e%5Ex%2B1%5Cright%29%5E%7B-2%2B1%7D%7D%7B-2%2B1%7D%3D3%5Ccdot%20-%5Cleft%28e%5Ex%2B1%5Cright%29%5E%7B-1%7D%3D-%5Cfrac%7B3%7D%7Be%5Ex%2B1%7D)
6. Add a constant to the solution
![-\frac{3}{e^x+1}+C](https://tex.z-dn.net/?f=-%5Cfrac%7B3%7D%7Be%5Ex%2B1%7D%2BC)
A = < 90°
the measure of the angle is less than 90
If she used 3 for 36 we can assume the muffin tins hold 12 muffins, so we can do 60 divided by 12 to get 5!
She uses 5 muffin tins!