find-dx-x-1-x-2-x-3-

Question Number 65355 by mathmax by abdo last updated on 28/Jul/19

f i n d \int \frac{d x}{\sqrt{(x + 1) (x + 2) (x + 3)}}

Commented by Prithwish sen last updated on 29/Jul/19

\int \frac{dx}{\sqrt{{{(x + 2)}^{2} - 1} (x + 2)}} putting (x + 2) = a

= \int \frac{da}{\sqrt{a (a^{2} - 1)}} again putting a = \sec θ

= \int \sqrt{\sec θ} d θ ? ? ? ?

Answered by ajfour last updated on 29/Jul/19

x + 2 = t

\int \frac{d x}{\sqrt{t (t^{2} - 1)}}

l e t t = \sec^{2} θ \Rightarrow d t = 2 \sec^{2} θ \tan θ d θ

\int \frac{2 \sec θ \tan θ d θ}{\sqrt{\sec^{4} θ - 1}}

l e t \sec θ = z

\int \frac{2 d z}{\sqrt{z^{4} - 1}} = ?

Answered by MJS last updated on 29/Jul/19

\int \frac{d x}{\sqrt{(x + 1) (x + 2) (x + 3)}} =

[t = arccosh (x + 2) \to d x = d t \sqrt{(x + 1) (x + 3)}]

= \int \frac{d t}{\sqrt{\cosh t}} =

[u = \frac{t}{2} i \to d t = - 2 i d u]

= - 2 i \int \frac{d u}{\sqrt{\cos 2 u}} = - 2 i \int \frac{d u}{\sqrt{1 - {2 \sin}^{2} u}} =

this is an incomplete elliptic integral

= - 2 i F (u ∣ 2)

\dots

Facebook Tweet Pin