Help With Solving This Differential Equation

helpmeimdumb

1. The problem statement, all variables and given/known data

1. The slope of a function at any point (x,y) is 〖2e〗^x/(e^x+2) . The point (0,2ln3) is on the graph of f.

(A) Write an equation of the tangent line to the graph of f at x=0.
(B) Use the tangent line in part A to approximate f(0.3) to the nearest thousandth.
(C) Solve the differential equation dy/(dx )=(2e^x)/(e^x+2) with the initial condition f(0)=2ln3.
(D) Use the solution in part C to find f(0.3) to the nearest thousandth.



2. Relevant equations



3. The attempt at a solution


(A)

dy/(dx )=(2e^x)/(e^x+2)

At x=0, dy/dx=(2e^0)/(e^0+2)=2/3

Equation of tangent line at x=0: y-2ln3=2/3 (x-0)

y-2ln3=2/3 x or y=2/3 x+2ln3

(B)

f(0.3)≈2/3 (0.3)+2ln3≈2.39722≈2.397

(C)

dy/dx=〖2e〗^x/(e^x+2) → dy=(2e^x)/(e^x+2) → ∫dy= ∫〖(2e^x)/(e^x+2) dx〗

Let u=e^x+2, du=e^x dx
2du=2e^x

Here is where I run into problems. Can anybody help me with the rest? Any help would be greatly appreciated.

helpmeimdumb

bump. please, i'm desperate for help.

SammyS

Rewrite your integral in terms of the variable, u .