solve-dx-c-b-ax-1-3-

Question Number 110245 by Her_Majesty last updated on 28/Aug/20

s o l v e \int \frac{d x}{\sqrt[3]{c - \sqrt{b - a x}}}

Commented by Lordose last updated on 28/Aug/20

\int \frac{dx}{\sqrt[3]{c - \sqrt{b - ax}}}

★ Solution

set u = b - ax

du = - adx

- \frac{1}{a} \int \frac{du}{\sqrt[3]{c - \sqrt{u}}}

set y = \sqrt{u}

du = 2 ydy

\frac{- 2}{a} \int \frac{ydy}{\sqrt[3]{c - y}}

set c - y = w

dw = - dy

\frac{2}{a} \int \frac{c - w}{\sqrt[3]{w}} dw

\frac{2 c}{a} w^{\frac{2}{3}} \times \frac{3}{2} - \frac{2}{a} w^{1 - \frac{1}{3} + 1} \times \frac{3}{5}

\frac{3 c}{a} w^{\frac{2}{3}} - \frac{6}{5 a} w^{\frac{5}{3}} + C

Commented by Her_Majesty last updated on 28/Aug/20

t h a n k y o u

Answered by Her_Majesty last updated on 28/Aug/20

i n o n e s t e p

t = \sqrt[3]{c - \sqrt{b - a x}} \Rightarrow d x = - \frac{6 t^{2} (t^{3} - c)}{a}

- \frac{6}{a} \int (t^{4} - c t) d t = \frac{3}{a} ({c t}^{2} - \frac{2 t^{5}}{5}) =

= \frac{3}{5 a} {(c - \sqrt{b - a x})}^{2 / 3} (3 c + 2 \sqrt{b - a x}) + C

Commented by bobhans last updated on 28/Aug/20

c o o l l \dots . g r e a t \dots n i c e

Commented by bobhans last updated on 28/Aug/20

b u t w h a t i s “ s “

Commented by Her_Majesty last updated on 28/Aug/20

t y p o s = a