find-x-2-9-x-3-dx-

Question Number 153535 by gsk2684 last updated on 08/Sep/21

f i n d \int \frac{\sqrt{x^{2} - 9}}{x^{3}} d x = ?

Answered by puissant last updated on 08/Sep/21

K = \int \frac{\sqrt{x^{2} - 9}}{x^{3}} d x

x = 3 s e c (u) \to d x = 3 s e c (u) t a n (u) d u

w e h a v e s e c (u) = \frac{x}{3} \Rightarrow c o s (u) = \frac{3}{x}

a n d s i n (u) = \frac{\sqrt{x^{2} - 9}}{x}

K = \int \frac{\sqrt{9 {s e c}^{2} u - 9}}{27 {s e c}^{3} u} \times 3 s e c (u) t a n (u) d u

= \int \frac{3 t a n (u)}{27 {s e c}^{2} u} \times 3 t a n (u) d u

= \frac{1}{3} \int {t a n}^{2} u {c o s}^{2} u d u

= \frac{1}{6} \int 2 {s i n}^{2} u d u = \frac{1}{6} \int (1 - c o s 2 u) d u

= \frac{1}{6} [u - \frac{s i n 2 u}{2}] + C

= \frac{1}{6} [u - s i n (u) c o s (u)] + C

\Rightarrow K = \frac{1}{6} [a r c c o s (\frac{3}{x}) - \frac{3}{x} \times \frac{\sqrt{x^{2} - 9}}{x}] + C

∴∵ K = \frac{1}{6} a r c c o s (\frac{3}{x}) - \frac{\sqrt{x^{2} - 9}}{2 x^{2}} + C

Commented by gsk2684 last updated on 13/Sep/21

t h a n k y o u

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