Question

Integrate the function ${\displaystyle \frac{x+2}{\sqrt{x^2+2x+3}}}$

Answer 1

Let ${\displaystyle I=\int \frac{x+2}{\sqrt{x^2+2x+3}} dx}$

${\displaystyle \frac{1}{2}\int \frac{2x+4}{\sqrt{x^2+2x+3}} dx}$

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

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

Let I = I₁ + I₂             ....(i)

Where ${\displaystyle I_1=\frac{1}{2}\int \frac{2x+2}{\sqrt{x^2+2x+3}} dx}$

Let x² + 2x + 3 = t

⇒ (2x + 2) dx = dt

∴ ${\displaystyle I_1=\frac{1}{2}\int \frac{dt}{\sqrt{t}}}$

${\displaystyle =\frac{1}{2}\int t^{-\frac{1}{2}}dt=\frac{1}{2}\times 2t^{\frac{1}{2}}}$

${\displaystyle =\frac{1}{2}\times 2\sqrt{x^2+2x+3}+C_1}$

${\displaystyle =\sqrt{x^2+2x+3}+C_1}$        .....(ii)

Also,

${\displaystyle I_2=\int \frac{dx}{\sqrt{x^2+2x+3}}}$

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

${\displaystyle =\frac{dx}{\sqrt{{(x+1)}^2+\sqrt{{\left(2\right)}^2}}}}$

${\displaystyle \log |(x+1)+\sqrt{{(x+1)}^2+2}|}$

${\displaystyle \log |(x+1)+\sqrt{x^2+2x+3}|+C_2}$      ....(iii)

Hence from (i), (ii) and (iii), we get

${\displaystyle I=\sqrt{x^2+2x+3}+\log |(x+1)+\sqrt{x^2+2x+3}|+C}$