find-dx-x-2-9-x-2-

Question Number 57749 by maxmathsup by imad last updated on 11/Apr/19

f i n d \int \frac{d x}{x^{2} \sqrt{9 + x^{2}}}

Commented by maxmathsup by imad last updated on 11/Apr/19

l e t A = \int \frac{d x}{x^{2} \sqrt{9 + x^{2}}} c h a n g e m e n t x = 3 s h (t) g i v e

A = \int \frac{3 c h (t)}{9 {s h}^{2} t 3 c h (t)} d t = \frac{1}{9} \int \frac{d t}{{s h}^{2} t} \Rightarrow 9 A = \int \frac{2 d t}{c h (2 t) - 1}

= \int \frac{2 d t}{\frac{e^{2 t} + e^{- 2 t}}{2} - 1} = 4 \int \frac{d t}{e^{2 t} + e^{- 2 t} - 2} =_{e^{2 t} = u} 4 \int \frac{d u}{2 u (u + u^{- 1} - 2)} (t = \frac{l n (u)}{2})

= 2 \int \frac{d u}{u^{2} + 1 - 2 u} = 2 \int \frac{d u}{{(u - 1)}^{2}} = \frac{- 2}{u - 1} + c = \frac{- 2}{e^{2 t} - 1} + c

w e h a v e s h (t) = \frac{x}{3} \Rightarrow t = a r g s h (\frac{x}{3}) = l n (\frac{x}{3} + \sqrt{1 + \frac{x^{2}}{9}}) \Rightarrow

e^{2 t} = {(\frac{x}{3} + \sqrt{1 + \frac{x^{2}}{9}})}^{2} = \frac{1}{9} {(x + \sqrt{x^{2} + 9})}^{2} \Rightarrow

9 A = \frac{2}{1 - \frac{1}{9} {(x + \sqrt{x^{2} + 9})}^{2}} + c = \frac{18}{9 - {(x + \sqrt{x^{2} + 9})}^{2}} + c \Rightarrow

A = \frac{2}{9 - {(x + \sqrt{x^{2} + 9})}^{2}} + C

Commented by MJS last updated on 11/Apr/19

\frac{2}{9 - {(x + \sqrt{x^{2} + 9})}^{2}} = \frac{2}{- 2 x^{2} - 2 x \sqrt{x^{2} + 9}} =

= - \frac{1}{x (x + \sqrt{x^{2} + 9})} = - \frac{x - \sqrt{x^{2} + 9}}{x (x + \sqrt{x^{2} + 9}) (x - \sqrt{x^{2} + 9})} =

= - \frac{x - \sqrt{x^{2} + 9}}{- 9 x} = \frac{1}{9} - \frac{\sqrt{x^{2} + 9}}{9 x}

A = - \frac{\sqrt{x^{2} + 9}}{9 x} + C

Commented by maxmathsup by imad last updated on 11/Apr/19

t h a n k s s i r f o r c o m p l e t i n g c a l c u l u s .

Answered by MJS last updated on 11/Apr/19

\frac{d}{d x} [\sqrt{x^{2} + 9}] = \frac{x}{\sqrt{x^{2} + 9}}

\frac{d}{d x} [\frac{\sqrt{x^{2} + 9}}{f (x)}] = \frac{x f (x) - (x^{2} + 9) f^{'} (x)}{\sqrt{x^{2} + 9} {(f (x))}^{2}}

trying f (x) = x this leads to - \frac{9}{x^{2} \sqrt{x^{2} + 9}}

trying f (x) = - 9 x this leads to \frac{1}{x^{2} \sqrt{x^{2} + 9}}

\Rightarrow \int \frac{d x}{x^{2} \sqrt{x^{2} + 9}} = - \frac{\sqrt{x^{2} + 9}}{9 x} + C

Facebook Tweet Pin