Question Number 149498 by peter frank last updated on 05/Aug/21
$$ \\ $$$$\int\frac{{px}+{q}}{\:\sqrt{{x}^{\mathrm{2}} +{r}^{\mathrm{2}} }} \\ $$
Answered by MJS_new last updated on 05/Aug/21
![∫((px+q)/( (√(x^2 +r^2 ))))dx= [t=((x+(√(x^2 +r^2 )))/r) → dx=((r(√(x^2 +r^2 )))/(x+(√(x^2 +r^2 ))))dt=((√(x^2 +r^2 ))/t)dt] =∫((prt^2 +2qt−pr)/(2t^2 ))dt=∫((q/t)−((pr)/(2t^2 ))+((pr)/2))dt= =qln t +((pr)/(2t))+((prt)/2)=qln t +((pr(t^2 +1))/(2t))= =qln (x+(√(x^2 +r^2 )))> +p(√(x^2 +r^2 )) +C](https://www.tinkutara.com/question/Q149507.png)
$$\int\frac{{px}+{q}}{\:\sqrt{{x}^{\mathrm{2}} +{r}^{\mathrm{2}} }}{dx}= \\ $$$$\:\:\:\:\:\left[{t}=\frac{{x}+\sqrt{{x}^{\mathrm{2}} +{r}^{\mathrm{2}} }}{{r}}\:\rightarrow\:{dx}=\frac{{r}\sqrt{{x}^{\mathrm{2}} +{r}^{\mathrm{2}} }}{{x}+\sqrt{{x}^{\mathrm{2}} +{r}^{\mathrm{2}} }}{dt}=\frac{\sqrt{{x}^{\mathrm{2}} +{r}^{\mathrm{2}} }}{{t}}{dt}\right] \\ $$$$=\int\frac{{prt}^{\mathrm{2}} +\mathrm{2}{qt}−{pr}}{\mathrm{2}{t}^{\mathrm{2}} }{dt}=\int\left(\frac{{q}}{{t}}−\frac{{pr}}{\mathrm{2}{t}^{\mathrm{2}} }+\frac{{pr}}{\mathrm{2}}\right){dt}= \\ $$$$={q}\mathrm{ln}\:{t}\:+\frac{{pr}}{\mathrm{2}{t}}+\frac{{prt}}{\mathrm{2}}={q}\mathrm{ln}\:{t}\:+\frac{{pr}\left({t}^{\mathrm{2}} +\mathrm{1}\right)}{\mathrm{2}{t}}= \\ $$$$={q}\mathrm{ln}\:\left({x}+\sqrt{{x}^{\mathrm{2}} +{r}^{\mathrm{2}} }\right)>\:+{p}\sqrt{{x}^{\mathrm{2}} +{r}^{\mathrm{2}} }\:+{C} \\ $$
Commented by peter frank last updated on 06/Aug/21
$${thank}\:{you}\: \\ $$
Answered by Ar Brandon last updated on 06/Aug/21
$${I}=\int\frac{{px}+{q}}{\:\sqrt{{x}^{\mathrm{2}} +{r}^{\mathrm{2}} }}{dx} \\ $$$$\:\:=\int\frac{{px}}{\:\sqrt{{x}^{\mathrm{2}} +{r}^{\mathrm{2}} }}{dx}+\int\frac{{q}}{\:\sqrt{{x}^{\mathrm{2}} +{r}^{\mathrm{2}} }}{dx} \\ $$$$\:\:=\frac{{p}}{\mathrm{2}}\int\frac{\mathrm{2}{x}}{\:\sqrt{{x}^{\mathrm{2}} +{r}^{\mathrm{2}} }}{dx}+\int\frac{{q}}{\:\sqrt{{x}^{\mathrm{2}} +{r}^{\mathrm{2}} }}{dx} \\ $$$$\:\:={p}\sqrt{{x}^{\mathrm{2}} +{r}^{\mathrm{2}} }+{q}\:{arcsinh}\left(\frac{{x}}{{r}}\right)+{C} \\ $$$$\:\:={p}\sqrt{{x}^{\mathrm{2}} +{r}^{\mathrm{2}} }+{q}\mathrm{ln}\left({x}+\sqrt{{x}^{\mathrm{2}} +{r}^{\mathrm{2}} }\right)+{C} \\ $$
Commented by peter frank last updated on 06/Aug/21
$${thank}\:{you} \\ $$