x-2-4-1-2-dx-trig-substitution-only-

Question Number 62251 by hovea cw last updated on 18/Jun/19

\int {(x^{2} - 4)}^{1 / 2} dx

trig substitution only

Commented by maxmathsup by imad last updated on 18/Jun/19

l e t I = \int \sqrt{x^{2} - 4} d x w e u s e t h e c h a n g e m e n t x = 2 c h (t) \Rightarrow

I = \int \sqrt{4 {c h}^{2} t - 4 (} 2 s h (t) d t = 4 \int \sqrt{{c h}^{2} t - 1} s h (t) d t = 4 \int {s h}^{2} t d t

= 4 \int \frac{c h (2 t) - 1}{2} d t = 2 \int c h (2 t) d t - 2 t = s h (2 t) - 2 t

= 2 s h (t) c h (t) - 2 t = x \sqrt{{c h}^{2} t - 1} - 2 t = x \sqrt{\frac{x^{2}}{4} - 1} - 2 t = \frac{x}{2} \sqrt{x^{2} - 4} - 2 a r g c h (\frac{x}{2})

= \frac{x}{2} \sqrt{x^{2} - 4} - 2 l n (\frac{x}{2} + \sqrt{\frac{x^{2}}{4} - 1}) = \frac{x}{2} \sqrt{x^{2} - 4} - 2 l n (\frac{x + \sqrt{x^{2} - 4}}{2}) + c_{0}

= \frac{x}{2} \sqrt{x^{2} - 4} - 2 l n (x + \sqrt{x^{2} - 4}) + 4 l n (2) + c_{0} \Rightarrow

\int \sqrt{x^{2} - 4} d x = \frac{x}{2} \sqrt{x^{2} - 4} - 2 l n (x + \sqrt{x^{2} - 4}) + C .