Hi, first I have to thank you guys for the help, I relly appreciate it.
I got two questions, and I have tried and got some results but I'm not sure if it's quite right or the methods im using are right.
Solve the volume of the rotating body that arise when the area between the curve y = 1/sqrt (x (1 + x ^ 2)) x> 1 and x axis, rotates about the x axis one revolution.
Y = 1/sqrt(x(1+x^2))
π 1/(x(1+x^2)) dx
Here I do intergration by the "chainrule"
π x^-1 * (1+x^2)^-1 dx =( [x^-1 * arctanx] + x^-2 * arctanx)π
I have trouble to find a primitive function to the last intergral so I do intergration one more time.
x^-2 * arctanx = [ x^1/-1 * arctanx] - x^-1 * (1+x^2)^-1
I add the two intergrations to each other
π x^-1 * (1+x^2)^-1 dx = [x^-1 * arctanx]+[x^-1 / -1 * arctan x] - x^-1*(1+x^2)^-1
2π x^-1 * (1+x^2)^-1 dx = [x^-1 * arctanx - x^-1 *arctanx] from 1 to positive eternity
π x^-1 * (1+x^2)^-1 dx = ([x^-1 * arctanx - x^-1 *arctanx])/2
π([x^-1 * arctanx - x^-1 *arctanx])/2 from 1 to positive eternity
Then I put in the two values, eternity and 1 and I get as result
π arctan 1
Question 2 Optimizing.
Diabetes can sometimes be treated by operating in a capsule that delivers insulin into the blood. A researcher examines a capsule type that produce insulin at a speed of 0:52 * t * e ^ (-0.89t) cm ^ 3/day where t is time in days. When the capsule emits at most per day and how much is it?
I start with doing som subsititons
e^(-0.89t) = u
-0.89t = lnu
t = lnu/(-0.89)
dt = 1/(u*-0,89) du
I put in the substitions
0.52 lnu/( -0.89) * u * (1/u * -0.89) du =
= 0.52 lnu /(0.7921) du =
(0.52/0.7921) 1 * ln u du = [ u * lnu] - u * (1/u)
(0.52/0.7921) 1 * ln u du = [ u * lnu] - [ u]
I reset the substition and set in the orginal values
(0.52/0.7921)( e^-0.89t * - 0.89t - e^-0.89t)
How and what should I do know to answer the questions?
When the capsule emits at most per day and how much is it?
Thanks for helping me