dx-x-3-x-2-a-2-

Question Number 118113 by bemath last updated on 15/Oct/20

\int \frac{d x}{x^{3} \sqrt{x^{2} - a^{2}}} = ?

Answered by Lordose last updated on 15/Oct/20

x = asecu dx = secutanudu

\frac{1}{a} \int \frac{secutanu}{\sec^{3} utanu} du

\frac{1}{a} \int \frac{1}{\sec^{2} u} du

\frac{1}{a} \int \cos^{2} udu

\frac{1}{a} (\frac{u}{2} + \frac{sinucosu}{2}) + C

secu = \frac{x}{a} \Rightarrow cosu = \frac{a}{x}

sinu = \sqrt{1 - \cos^{2} u} = \sqrt{1 - {(\frac{a}{x})}^{2}}

\frac{1}{2 a} (\sec^{- 1} (\frac{x}{a}) + \frac{a \sqrt{x^{2} - a^{2}}}{x^{2}}) + C

Commented by bemath last updated on 15/Oct/20

y e s \dots t h a n k y o u

Answered by 1549442205PVT last updated on 15/Oct/20

Put \sqrt{x^{2} - a^{2}} = t \Rightarrow dt = \frac{xdx}{\sqrt{x^{2} - a^{2}}}

dx = tdt / x

\int \frac{d x}{x^{3} \sqrt{x^{2} - a^{2}}} = \int \frac{tdt}{x^{4} . t} = \int \frac{dt}{{(t^{2} + a^{2})}^{2}}

I_{2} = \int \frac{dt}{{(t^{2} + a^{2})}^{2}} . Put t = atan φ

\Rightarrow dt = a (1 + \tan^{2} φ) d φ = a (1 + t^{2}) d φ

I_{2} = \int \frac{a (1 + \tan^{2} φ) d φ}{a^{4} {(\tan^{2} φ + 1)}^{2}} = \int \frac{d φ}{a^{3} (1 + \tan^{2} φ)}

= \frac{1}{a^{3}} \int \cos^{2} φ d φ = \frac{1}{{2 a}^{3}} \int (1 + \cos 2 φ) d φ

= \frac{φ}{{2 a}^{3}} + \frac{1}{{4 a}^{3}} \sin 2 φ = \frac{\tan^{- 1} (\frac{\sqrt{x^{2} - a^{2}}}{a})}{{2 a}^{3}}

+ \frac{1}{{4 a}^{3}} \times \frac{2 a \sqrt{x^{2} - a^{2}}}{x^{2}}

= \frac{1}{{2 a}^{3}} [\tan^{- 1} (\frac{\sqrt{x^{2} - a^{2}}}{a}) + \frac{a \sqrt{x^{2} - a^{2}}}{x^{2}}] + C

Answered by TANMAY PANACEA last updated on 15/Oct/20

t^{2} = x^{2} - a^{2}

\int \frac{x d x}{x^{4} \sqrt{x^{2} - a^{2}}}

\int \frac{t d t}{{(t^{2} + a^{2})}^{2} t} \to t = a t a n α

\int \frac{{s e c}^{2} α d α}{a^{4} {s e c}^{4} α} = \frac{1}{a^{2}} \int \frac{1 + c o s 2 α}{2} d α

\frac{1}{a^{2}} \times \frac{1}{2} (α + \frac{s i n 2 α}{2}) + C

\frac{1}{2 a^{2}} ({t a n}^{-} (\frac{t}{a}) + \frac{1}{2} \times \frac{2 t a n α}{1 + {t a n}^{2} α})

= \frac{1}{2 a^{2}} {t a n}^{- 1} (\frac{\sqrt{x^{2} - a^{2}}}{a}) + \frac{1}{2 a^{2}} (\frac{\frac{t}{a}}{1 + \frac{t^{2}}{a^{2}}}) + C

= \frac{1}{2 a^{2}} \times {t a n}^{- 1} (\frac{\sqrt{x^{2} - a^{2}}}{a}) + \frac{1}{2 a^{2}} (\frac{\sqrt{x^{2} - a^{2}}}{a} \times \frac{a^{2}}{x^{2}}) + c

= \frac{1}{2 a^{2}} {t a n}^{- 1} (\frac{\sqrt{x^{2} - a^{2}}}{a}) + \frac{1}{2 a^{2}} (\frac{a \sqrt{x^{2} - a^{2}}}{x^{2}}) + c