square root integral

intervade

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Apr 6, 2009
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Hello,

I'm having a problem with this integral. Its almost like it could be a basic u, du substitution but I'm not really sure where to start.

1x31+x4dx\displaystyle \int \frac{1}{x^3\sqrt{1+x^4}}dx

Some direction on this would be much appreciated!
 
dxx3(1+x4)1/2, Let u = (1+x4)1/2, du = 12(1+x4)1/2(4x3)dx\displaystyle \int\frac{dx}{x^{3}(1+x^{4})^{1/2}}, \ Let \ u \ = \ (1+x^{4})^{1/2}, \ du \ = \ \frac{1}{2}(1+x^{4})^{-1/2}(4x^{3})dx

du = 2x3udx, dx = udu2x3\displaystyle du \ = \ \frac{2x^{3}}{u}dx, \ dx \ = \ \frac{udu}{2x^{3}}

Also note, 1+x4 = u2, x4 = u21, x6 = (u21)3/2\displaystyle Also \ note, \ 1+x^{4} \ = \ u^{2}, \ x^{4} \ = \ u^{2}-1, \ x^{6} \ = \ (u^{2}-1)^{3/2}

Hence, 12udux6u = 12dux6 = 12du(u21)3/2 =  u2(u21)1/2 +C\displaystyle Hence, \ \frac{1}{2}\int\frac{udu}{x^{6}u} \ = \ \frac{1}{2}\int\frac{du}{x^{6}} \ = \ \frac{1}{2}\int\frac{du}{(u^{2}-1)^{3/2}} \ = \ \ - \frac{u}{2(u^{2}-1)^{1/2}} \ +C

ReSubbing, we get =  (1+x4)1/22(x4)1/2+C = (1+x4)1/22x2+C.\displaystyle Re- Subbing, \ we \ get \ = \ - \ \frac{(1+x^{4})^{1/2}}{2(x^{4})^{1/2}}+C \ = \ -\frac{(1+x^{4})^{1/2}}{2x^{2}}+C.

Check: Dx[(1+x4)1/22x2+C] = 1x3(1+x4)1/2.\displaystyle Check: \ D_x\bigg[-\frac{(1+x^{4})^{1/2}}{2x^{2}}+C\bigg] \ = \ \frac{1}{x^{3}(1+x^{4})^{1/2}}.
 
This is probably the easiest way, to wit:\displaystyle This \ is \ probably \ the \ easiest \ way, \ to \ wit:

dxx3(1+x4)1/2 = dxx3[1+(x2)2]1/2\displaystyle \int\frac{dx}{x^{3}(1+x^{4})^{1/2}} \ = \ \int\frac{dx}{x^{3}[1+(x^{2})^{2}]^{1/2}}

Now, let x2 = tan(θ),      x = tan1/2(θ) and x3 = tan3/2(θ)\displaystyle Now, \ let \ x^{2} \ = \ tan(\theta), \ \implies \ x \ = \ tan^{1/2}(\theta) \ and \ x^{3} \ = \ tan^{3/2}(\theta)

Also, 2xdx = sec2(θ)dθ, dx = sec2(θ)dθ2x = sec2(θ)dθ2tan1/2(θ)\displaystyle Also, \ 2xdx \ = \ sec^{2}(\theta)d\theta, \ dx \ = \ \frac{sec^{2}(\theta)d\theta}{2x} \ = \ \frac{sec^{2}(\theta)d\theta}{2tan^{1/2}(\theta)}

Hence, we have: dxx3[1+(x2)2]1/2 = 12sec2(θ)dθtan1/2(θ)tan3/2(θ)[1+tan2(θ)]1/2\displaystyle Hence, \ we \ have: \ \int\frac{dx}{x^{3}[1+(x^{2})^{2}]^{1/2}} \ = \ \frac{1}{2}\int\frac{sec^{2}(\theta)d\theta}{tan^{1/2}(\theta)tan^{3/2}(\theta)[1+tan^{2}(\theta)]^{1/2}}

= 12sec2(θ)dθtan2(θ)sec(θ) = 12sec(θ)dθtan2(θ) = 12cos(θ)dθsin2(θ),\displaystyle = \ \frac{1}{2}\int\frac{sec^{2}(\theta)d\theta}{tan^{2}(\theta)sec(\theta)} \ = \ \frac{1}{2}\int\frac{sec(\theta)d\theta}{tan^{2}(\theta)} \ = \ \frac{1}{2}\int\frac{cos(\theta)d\theta}{sin^{2}(\theta)},

= 12cot(θ)csc(θ)dθ = 12csc(θ)+C = 12[(x4+1)1/2x2]+C = (x4+1)1/22x2+C\displaystyle = \ \frac{1}{2}\int cot(\theta)csc(\theta)d\theta \ = \ -\frac{1}{2}csc(\theta)+C \ = \ -\frac{1}{2}\bigg[\frac{(x^{4}+1)^{1/2}}{x^{2}}\bigg]+C \ = \ -\frac{(x^{4}+1)^{1/2}}{2x^{2}}+C
 
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