- #1

- 315

- 2

**1. The problem, the whole problem, and nothing but the problem**

[tex] \int \frac{dx}{ \sqrt{ 1+ \sqrt{ 1+ \sqrt{ x } } } } [/tex]

## Homework Equations

u-substitution (in the style of trig substitution)

I think that I've got it figured out, I just don't know if my substitutions were legitimate.

## The Attempt at a Solution

[tex] \int \frac{dx}{ \sqrt{ 1+ \sqrt{ 1+ \sqrt{ x } } } } [/tex]

[tex] u^2 = x, 2u \, du = dx [/tex]

[tex] 2 \int \frac{u}{ \sqrt{ 1+ \sqrt{ 1+ u } } } du [/tex]

[tex] v^2 = 1+u, 2v \, dv = du [/tex]

[tex] 4 \int \frac{v(v^2 - 1)}{ \sqrt{ 1+ v } } dv [/tex]

Now just simplifying:

[tex] 4 \int \frac{v(v+1)(v-1)\sqrt{ 1+ v }}{ v+1 } dv [/tex]

[tex] 4 \int v(v-1)\sqrt{ 1+ v } \, dv [/tex]

[tex] w^2 = 1+v, 2w \, dw = dv [/tex]

[tex] 8 \int w(w^2 -1 )(w^2 - 2)(w) \, dw [/tex]

[tex] 8 \int w^2(w^2 -1 )(w^2 - 2) \, dw [/tex]

I'm sure that I could foil this out, integrate, and make the 4 or 5 substitutions. Before I do so, is my work justified? If so, is there a trick to the last integral (without foiling it all out)?

Thanks