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Question Number 149498 by peter frank last updated on 05/Aug/21

\int \frac{p x + q}{\sqrt{x^{2} + r^{2}}}

Answered by MJS_new last updated on 05/Aug/21

\int \frac{p x + q}{\sqrt{x^{2} + r^{2}}} d x =

[t = \frac{x + \sqrt{x^{2} + r^{2}}}{r} \to d x = \frac{r \sqrt{x^{2} + r^{2}}}{x + \sqrt{x^{2} + r^{2}}} d t = \frac{\sqrt{x^{2} + r^{2}}}{t} d t]

= \int \frac{{p r t}^{2} + 2 q t - p r}{2 t^{2}} d t = \int (\frac{q}{t} - \frac{p r}{2 t^{2}} + \frac{p r}{2}) d t =

= q \ln t + \frac{p r}{2 t} + \frac{p r t}{2} = q \ln t + \frac{p r (t^{2} + 1)}{2 t} =

= q \ln (x + \sqrt{x^{2} + r^{2}}) > + p \sqrt{x^{2} + r^{2}} + C

Commented by peter frank last updated on 06/Aug/21

t h a n k y o u

Answered by Ar Brandon last updated on 06/Aug/21

I = \int \frac{p x + q}{\sqrt{x^{2} + r^{2}}} d x

= \int \frac{p x}{\sqrt{x^{2} + r^{2}}} d x + \int \frac{q}{\sqrt{x^{2} + r^{2}}} d x

= \frac{p}{2} \int \frac{2 x}{\sqrt{x^{2} + r^{2}}} d x + \int \frac{q}{\sqrt{x^{2} + r^{2}}} d x

= p \sqrt{x^{2} + r^{2}} + q a r c s i n h (\frac{x}{r}) + C

= p \sqrt{x^{2} + r^{2}} + q \ln (x + \sqrt{x^{2} + r^{2}}) + C

Commented by peter frank last updated on 06/Aug/21

t h a n k y o u