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.

Question 1

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 y^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?

0.52te^(-0.89t) dt

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