Question

Solve

$\int \frac{x+2}{\sqrt{x^2+2x+3}}dx$

Answer 1

$\int \frac{x+2}{\sqrt{x^2+2x+3}}dx$

$=\int \frac{x+1}{\sqrt{x^2+2x+3}}dx+\int \frac{1}{\sqrt{x^2+2x+3}}dx$

Let $x^2+2x+3=t$

$=2x+2=\frac{dt}{dx}$

$=2(x+1)=\frac{dt}{dx}$

$=(x+1)dx=\frac{dt}{2}$

$=\int \frac{dt}{2\sqrt{t}}+\int \frac{1}{\sqrt{x^2+2x+1-1+3}}dx$

$=\frac{1}{2}\left[\frac{t^{-1/2+1}}{-\frac{1}{2}+1}\right]+\int \frac{1}{\sqrt{{(x+1)}^2+{\left(\sqrt{2}\right)}^2}}dx$

$=\frac{1}{2}\frac{t^{1/2}}{\frac{1}{2}}+\log (x+1+\sqrt{x^2+2x+3})$

$=t^{1/2}+\log (x+1+\sqrt{x^2+2x+3})$

$=\sqrt{x^2+2x+3}+\log (x+1+\sqrt{x^2+2x+3})+c$

Hence, the answer is $\sqrt{x^2+2x+3}+\log (x+1+\sqrt{x^2+2x+3})+c.$