- #1
Knissp
- 75
- 0
Homework Statement
[tex]\int_0^1 (6t^2 (1+9t^2)^{1/2} dt) [/tex]
Homework Equations
[tex]\int u dv = u v - \int v du [/tex]
The Attempt at a Solution
[tex]\int_0^1 (6t^2 (1+9t^2)^{1/2} dt) [/tex]
[tex]=6 * \int (t^2 (1+9t^2)^{1/2} dt)[/tex]
[tex]= 6 * \int (t * t (1+9t^2)^{1/2} dt)[/tex]
Let [tex]u = t [/tex]; let [tex]dv = t (1+9t^2)^{1/2} dt [/tex];
then [tex]du = dt[/tex]; and [tex]v = \int t (1+9t^2)^{1/2} dt[/tex]
(using w-substitution:
[tex] w = 1+9t^2[/tex],
[tex] dw = 18t dt[/tex];
[tex] dw/18=dt[/tex];
[tex] \int t (1+9t^2)^{1/2} dt [/tex]
=[tex] 1/18 \int w^{1/2} dw = 1/18 * 2/3 w^{3/2} = w^{3/2} / 27 = (1+9t^2)^{3/2}/27 [/tex]
[tex] v = [(1+9t^2)^{3/2}/27 [/tex]
[tex] \int(u dv) = u v - \int (v du) [/tex]
[tex] = t * (1+9t^2)^{3/2}/27 - \int ((1+9t^2)^{3/2}/27 dt)[/tex]
now i need help integrating [tex] (1+9t^2)^{3/2}/27 [/tex].