Answer:
Geometrical sequence of x2, each number is being multiplied by 2.
Step-by-step explanation:
10/5 = 2
20/10 = 2
40/20 = 2
Answer:
50 degrees
Step-by-step explanation:
The 40⁰ angle is bisecting a right, or 90⁰ angle. The total of angles f and 40⁰ = 90⁰
f + 40⁰ = 90⁰
f = 50⁰
Answer:
![(\frac{(1+e^{2x}) ^{\frac{5}{2} } }{{5}} + \frac{(1+e^{2x} )^{\frac{3}{2} } }{{3}} )+C](https://tex.z-dn.net/?f=%28%5Cfrac%7B%281%2Be%5E%7B2x%7D%29%20%5E%7B%5Cfrac%7B5%7D%7B2%7D%20%7D%20%7D%7B%7B5%7D%7D%20%2B%20%5Cfrac%7B%281%2Be%5E%7B2x%7D%20%29%5E%7B%5Cfrac%7B3%7D%7B2%7D%20%7D%20%7D%7B%7B3%7D%7D%20%29%2BC)
Step-by-step explanation:
<u><em> Step(i):-</em></u>
Given that the function
f(x) = ![e^{4x} \sqrt{1+e^{2x} }](https://tex.z-dn.net/?f=e%5E%7B4x%7D%20%5Csqrt%7B1%2Be%5E%7B2x%7D%20%7D)
Now integrating on both sides, we get
![\int\limits{f(x)} \, dx = \int\limits{e^{4x} \sqrt{1+e^{2x} } dx](https://tex.z-dn.net/?f=%5Cint%5Climits%7Bf%28x%29%7D%20%5C%2C%20dx%20%3D%20%5Cint%5Climits%7Be%5E%7B4x%7D%20%5Csqrt%7B1%2Be%5E%7B2x%7D%20%7D%20dx)
= ![\int\limits{e^{2x} e^{2x} \sqrt{1+e^{2x} } dx](https://tex.z-dn.net/?f=%5Cint%5Climits%7Be%5E%7B2x%7D%20e%5E%7B2x%7D%20%5Csqrt%7B1%2Be%5E%7B2x%7D%20%7D%20dx)
<u><em>Step(ii):-</em></u>
Let ![1 + e^{2x} = t](https://tex.z-dn.net/?f=1%20%2B%20e%5E%7B2x%7D%20%20%3D%20t)
![2e^{2x}dx = d t](https://tex.z-dn.net/?f=2e%5E%7B2x%7Ddx%20%3D%20d%20t)
![e^{2x}dx = \frac{1}{2} d t](https://tex.z-dn.net/?f=e%5E%7B2x%7Ddx%20%3D%20%5Cfrac%7B1%7D%7B2%7D%20d%20t)
= ![\int\limits{( \sqrt{1+e^{2x} }) e^{2x} e^{2x} dx](https://tex.z-dn.net/?f=%5Cint%5Climits%7B%28%20%5Csqrt%7B1%2Be%5E%7B2x%7D%20%7D%29%20e%5E%7B2x%7D%20e%5E%7B2x%7D%20dx)
=
= ![\frac{1}{2} \int\limits {\sqrt{t} (t) -\sqrt{t} ) dt }](https://tex.z-dn.net/?f=%5Cfrac%7B1%7D%7B2%7D%20%5Cint%5Climits%20%7B%5Csqrt%7Bt%7D%20%28t%29%20-%5Csqrt%7Bt%7D%20%29%20dt%20%7D)
= ![\frac{1}{2} \int\limits {(t^{\frac{1}{2} } t^{1} +t^{\frac{1}{2} } ) } \, dx](https://tex.z-dn.net/?f=%5Cfrac%7B1%7D%7B2%7D%20%5Cint%5Climits%20%7B%28t%5E%7B%5Cfrac%7B1%7D%7B2%7D%20%20%7D%20t%5E%7B1%7D%20%2Bt%5E%7B%5Cfrac%7B1%7D%7B2%7D%20%7D%20%29%20%7D%20%5C%2C%20dx)
![= \frac{1}{2} \int\limits {(t^{\frac{3}{2} } +t^{\frac{1}{2} } ) } \, dx](https://tex.z-dn.net/?f=%3D%20%5Cfrac%7B1%7D%7B2%7D%20%5Cint%5Climits%20%7B%28t%5E%7B%5Cfrac%7B3%7D%7B2%7D%20%20%7D%20%2Bt%5E%7B%5Cfrac%7B1%7D%7B2%7D%20%7D%20%29%20%7D%20%5C%2C%20dx)
= ![\frac{1}{2} (\frac{t^{\frac{3}{2} +1} }{\frac{3}{2}+1 } + \frac{t^{\frac{1}{2} +1} }{\frac{1}{2}+1 } )+C](https://tex.z-dn.net/?f=%5Cfrac%7B1%7D%7B2%7D%20%28%5Cfrac%7Bt%5E%7B%5Cfrac%7B3%7D%7B2%7D%20%2B1%7D%20%7D%7B%5Cfrac%7B3%7D%7B2%7D%2B1%20%7D%20%2B%20%5Cfrac%7Bt%5E%7B%5Cfrac%7B1%7D%7B2%7D%20%2B1%7D%20%7D%7B%5Cfrac%7B1%7D%7B2%7D%2B1%20%7D%20%29%2BC)
= ![\frac{1}{2} (\frac{t^{\frac{3}{2} +1} }{\frac{5}{2} } + \frac{t^{\frac{1}{2} +1} }{\frac{3}{2} } )+C](https://tex.z-dn.net/?f=%5Cfrac%7B1%7D%7B2%7D%20%28%5Cfrac%7Bt%5E%7B%5Cfrac%7B3%7D%7B2%7D%20%2B1%7D%20%7D%7B%5Cfrac%7B5%7D%7B2%7D%20%7D%20%2B%20%5Cfrac%7Bt%5E%7B%5Cfrac%7B1%7D%7B2%7D%20%2B1%7D%20%7D%7B%5Cfrac%7B3%7D%7B2%7D%20%7D%20%29%2BC)
= ![\frac{1}{2} (\frac{t^{\frac{5}{2} } }{\frac{5}{2} } + \frac{t^{\frac{3}{2} } }{\frac{3}{2} } )+C](https://tex.z-dn.net/?f=%5Cfrac%7B1%7D%7B2%7D%20%28%5Cfrac%7Bt%5E%7B%5Cfrac%7B5%7D%7B2%7D%20%7D%20%7D%7B%5Cfrac%7B5%7D%7B2%7D%20%7D%20%2B%20%5Cfrac%7Bt%5E%7B%5Cfrac%7B3%7D%7B2%7D%20%7D%20%7D%7B%5Cfrac%7B3%7D%7B2%7D%20%7D%20%29%2BC)
= ![(\frac{(1+e^{2x}) ^{\frac{5}{2} } }{{5}} + \frac{(1+e^{2x} )^{\frac{3}{2} } }{{3}} )+C](https://tex.z-dn.net/?f=%28%5Cfrac%7B%281%2Be%5E%7B2x%7D%29%20%5E%7B%5Cfrac%7B5%7D%7B2%7D%20%7D%20%7D%7B%7B5%7D%7D%20%2B%20%5Cfrac%7B%281%2Be%5E%7B2x%7D%20%29%5E%7B%5Cfrac%7B3%7D%7B2%7D%20%7D%20%7D%7B%7B3%7D%7D%20%29%2BC)
<u><em>Final answer:-</em></u>
= ![(\frac{(1+e^{2x}) ^{\frac{5}{2} } }{{5}} + \frac{(1+e^{2x} )^{\frac{3}{2} } }{{3}} )+C](https://tex.z-dn.net/?f=%28%5Cfrac%7B%281%2Be%5E%7B2x%7D%29%20%5E%7B%5Cfrac%7B5%7D%7B2%7D%20%7D%20%7D%7B%7B5%7D%7D%20%2B%20%5Cfrac%7B%281%2Be%5E%7B2x%7D%20%29%5E%7B%5Cfrac%7B3%7D%7B2%7D%20%7D%20%7D%7B%7B3%7D%7D%20%29%2BC)
Answer: You would go down to -4 on the y axis. Then horizontal make a line and shade everything underneath of it.
Step-by-step explanation: