Calculate-9-x-2-x-6-dx-

Question Number 76892 by john santu last updated on 31/Dec/19

C a l c u l a t e \int \frac{\sqrt{9 - x^{2}}}{x^{6}} d x .

Answered by jagoll last updated on 31/Dec/19

Answered by MJS last updated on 01/Jan/20

\int \frac{\sqrt{9 - x^{2}}}{x^{6}} d x =

[t = \frac{x}{\sqrt{9 - x^{2}}} \to d x = \frac{{(9 - x^{2})}^{\frac{3}{2}}}{9} d t]

= \frac{1}{81} \int \frac{d t}{t^{4}} + \frac{1}{81} \int \frac{d t}{t^{6}} = - \frac{1}{243 t^{3}} - \frac{1}{405 t^{5}} =

= - \frac{{(9 - x^{2})}^{\frac{3}{2}}}{243 x^{3}} - \frac{{(9 - x^{2})}^{\frac{5}{2}}}{405 x^{5}} = - \frac{(2 x^{2} + 27) {(9 - x^{2})}^{\frac{3}{2}}}{1215 x^{5}} + C

Commented by jagoll last updated on 01/Jan/20

s o r r y s i r . t h e e q u a t i o n \int \frac{\sqrt{9 - x^{2}}}{x^{6}} d x .

Commented by MJS last updated on 01/Jan/20

thank you, {it}^{'} s just a typo, I corrected it

Commented by jagoll last updated on 01/Jan/20

t h a n k s y o u s i r

Answered by petrochengula last updated on 03/Jan/20

l e t x = 3 s i n θ \Rightarrow d x = 3 c o s θ d θ

= \int \frac{\sqrt{9 - 9 {s i n}^{2} θ}}{3^{6} {s i n}^{6} θ} 3 c o s θ d θ = \frac{1}{3^{4}} \int \frac{{c o s}^{2} θ}{{s i n}^{6} θ} d θ

= \frac{1}{81} \int {c o t}^{2} θ {c o s e c}^{4} θ d θ

= \frac{1}{81} \int {c o t}^{2} θ (1 + {c o t}^{2} θ) {c o s e c}^{2} θ d θ

l e t t = c o t θ \Rightarrow - d t = {c o s e c}^{2} θ d θ

= - \frac{1}{81} \int t^{2} (1 + t^{2}) d t

= - \frac{t^{3}}{243} - \frac{t^{5}}{405} + c

= - \frac{{c o t}^{3} θ}{243} - \frac{{c o t}^{5} θ}{405} + c

= - \frac{{(c o t ({s i n}^{- 1} (\frac{x}{3})))}^{3}}{243} - \frac{{(c o t ({s i n}^{- 1} (\frac{x}{3})))}^{5}}{405} + c