1-1-t-2-t-dt-

Question Number 206248 by MetaLahor1999 last updated on 10/Apr/24

\int \frac{1}{\sqrt{(1 - t) (2 - t)}} d t = \dots ?

Answered by TonyCWX08 last updated on 10/Apr/24

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

= \int \frac{1}{\sqrt{{(t - \frac{3}{2})}^{2} - \frac{1}{4}}} d t

l e t

u = t - \frac{3}{2}

d u = d x

\int \frac{d u}{\sqrt{u^{2} - \frac{1}{4}}}

= l n (∣ u + \sqrt{u^{2} - \frac{1}{4}} ∣) + c

= l n (∣ t - \frac{3}{2} + \sqrt{{(t - \frac{3}{2})}^{2} - \frac{1}{4}} ∣) + c

= l n (∣ t - \frac{3}{2} + \sqrt{t^{2} - 3 t + 2} ∣) + c

Commented by Frix last updated on 10/Apr/24

Yes . But for the less advanced it would be

nice to show the path for

\int \frac{d u}{\sqrt{u^{2} \pm c}} = \ln ∣ u + \sqrt{u^{2} \pm c} ∣

Commented by TonyCWX08 last updated on 11/Apr/24

I f y o u w a n t, s u r e .

B u t i t i n v o l v e s s o m e t r i g o n o m e t r y s u b s t i t u t i o n .