Answer:
Yes, this is a translation which doesn't change the pre-image size
Step-by-step explanation:
Answer:
The solution is ![\frac{1}{10} * tan^{-1}[\frac{e^{2x}}{5} ] + C](https://tex.z-dn.net/?f=%5Cfrac%7B1%7D%7B10%7D%20%2A%20tan%5E%7B-1%7D%5B%5Cfrac%7Be%5E%7B2x%7D%7D%7B5%7D%20%5D%20%2B%20%20C)
Step-by-step explanation:
From the question
The function given is ![f(x) = \frac{e^{2x}}{ 25 + e^{4x}} dx](https://tex.z-dn.net/?f=f%28x%29%20%3D%20%20%5Cfrac%7Be%5E%7B2x%7D%7D%7B%2025%20%2B%20e%5E%7B4x%7D%7D%20dx)
The indefinite integral is mathematically represented as
![\int\limits {\frac{e^{2x}}{ 25 + e^{4x}}} \, dx](https://tex.z-dn.net/?f=%5Cint%5Climits%20%20%7B%5Cfrac%7Be%5E%7B2x%7D%7D%7B%2025%20%2B%20e%5E%7B4x%7D%7D%7D%20%5C%2C%20dx)
Now let ![e^{2x} = u](https://tex.z-dn.net/?f=e%5E%7B2x%7D%20%3D%20%20u)
=> ![\frac{du}{dx} 2e^{2x}](https://tex.z-dn.net/?f=%5Cfrac%7Bdu%7D%7Bdx%7D%202e%5E%7B2x%7D)
=> ![2 e^{2x}dx = du](https://tex.z-dn.net/?f=2%20e%5E%7B2x%7Ddx%20%3D%20%20du%20)
So
![\int\limits {\frac{e^{2x}}{ 25 + e^{4x}}} \, dx = \int\limits {\frac{1}{ 2(25 + u^2)} } \, du](https://tex.z-dn.net/?f=%20%5Cint%5Climits%20%20%7B%5Cfrac%7Be%5E%7B2x%7D%7D%7B%2025%20%2B%20e%5E%7B4x%7D%7D%7D%20%5C%2C%20dx%20%3D%20%20%5Cint%5Climits%20%20%7B%5Cfrac%7B1%7D%7B%202%2825%20%2B%20u%5E2%29%7D%20%7D%20%5C%2C%20du)
![= \frac{1}{2} \frac{tan^{-1} [\frac{u}{5} ]}{5} + C](https://tex.z-dn.net/?f=%3D%20%5Cfrac%7B1%7D%7B2%7D%20%5Cfrac%7Btan%5E%7B-1%7D%20%5B%5Cfrac%7Bu%7D%7B5%7D%20%5D%7D%7B5%7D%20%20%2B%20%20C)
Now substituting for u
![\frac{1}{10} * tan^{-1}[\frac{e^{2x}}{5} ] + C](https://tex.z-dn.net/?f=%5Cfrac%7B1%7D%7B10%7D%20%2A%20tan%5E%7B-1%7D%5B%5Cfrac%7Be%5E%7B2x%7D%7D%7B5%7D%20%5D%20%2B%20%20C)
Both of these problems will be solved in a similar way, but with different numbers. First, we set up an equation with the values given. Then, we solve. Lastly, we plug into the original expressions to solve for the angles.
[23] ABD = 42°, DBC = 35°
(4x - 2) + (3x + 2) = 77°
4x+ 3x + 2 - 2 = 77°
4x+ 3x= 77°
7x= 77°
x= 11°
-
ABD = (4x - 2) = (4(11°) - 2) = 44° - 2 = 42°
DBC = (3x + 2) = (3(11°) + 2) = 33° + 2 = 35°
[24] ABD = 62°, DBC = 78°
(4x - 8) + (4x + 8) = 140°
4x + 4x + 8 - 8 = 140°
4x + 4x = 140°
8x = 140°
8x = 140°
x = 17.5°
-
ABD = (4x - 8) = (4(17.5°) - 8) = 70° - 8° = 62°
DBC =(4x + 8) = (4(17.5°) + 8) = 70° + 8° = 78°
337.5 divide 450 by 4 and multiply the product by 3
2/3+(-1/3)
2/3-1/3
1/3
Your answer is 1/3