Mathematics - Indefinite Integration Question with Solution | TestHub

$\int \sqrt{x+\sqrt{x^2+2}}dx$ $\sqrt{x+\sqrt{x^2+2}}=t\Rightarrow x+\sqrt{x^2+2}=t^2$ $\sqrt{x^2+2}=t^2-x$ $\Rightarrow x^2+2=t^4+x^2-2t^{2}x$ $\Rightarrow \mathrm{x}=\frac{{\mathrm{t}}^4-2}{2{\mathrm{t}}^2}\Rightarrow \mathrm{dx}=\frac{{\mathrm{t}}^4+2}{{\mathrm{t}}^3}\mathrm{dt}$ $\int \mathrm{t}\cdot \frac{{\mathrm{t}}^4+2}{{\mathrm{t}}^3}\mathrm{dt}=\int \left({\mathrm{t}}^2+\frac{2}{{\mathrm{t}}^2}\right)\mathrm{dt}=\frac{{\mathrm{t}}^3}{3}-\frac{2}{\mathrm{t}}+\mathrm{C}$ $=\frac{t^3}{3}-\frac{2}{t}+C=\frac{t^4-6}{3t}+C$ $=\frac{{\left(x+\sqrt{x^2+2}\right)}^2-6}{3\sqrt{x+\sqrt{x^2+2}}}+C$