find-dx-x-2-x-2-x-7-

Question Number 73483 by abdomathmax last updated on 13/Nov/19

f i n d \int \frac{d x}{x + 2 - \sqrt{x^{2} - x + 7}}

Answered by MJS last updated on 13/Nov/19

\int \frac{d x}{x + 2 - \sqrt{x^{2} - x + 7}} = \int \frac{d x}{x + 2 - \sqrt{{(x - \frac{1}{2})}^{2} + \frac{27}{4}}} =

[t = \frac{\sqrt{3}}{9} (2 x - 1) \to d x = \frac{3 \sqrt{3}}{2} d t]

= \int \frac{d t}{t + \frac{5 \sqrt{3}}{9} - \sqrt{t^{2} + 1}} =

[t = \sinh \ln u = \frac{u^{2} - 1}{2 u} \Rightarrow u = t + \sqrt{t^{2} + 1} \to d t = \frac{u^{2} + 1}{2 u^{2}} d u]

= \frac{3 \sqrt{3}}{2} \int \frac{u^{2} + 1}{u (5 u - 3 \sqrt{3})} d u =

= \frac{3 \sqrt{3}}{10} \int d u - \frac{1}{2} \int \frac{d u}{u} + \frac{26}{25} \int \frac{d u}{u - \frac{3 \sqrt{3}}{5}} =

= \frac{3 \sqrt{3}}{10} u - \frac{1}{2} \ln u + \frac{26}{25} \ln (5 u - 3 \sqrt{3}) =

\dots

= \frac{2 x - 1 + 2 \sqrt{x^{2} - x + 7}}{10} - \frac{1}{2} \ln (2 x - 1 + 2 \sqrt{x^{2} - x + 7}) + \frac{26}{25} \ln (5 x - 16 + 5 \sqrt{x^{2} - x + 7}) + C

Commented by abdomathmax last updated on 17/Nov/19

t h a n k s s i r m j s .