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Question Number 88010 by Chi Mes Try last updated on 07/Apr/20

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

Commented by niroj last updated on 07/Apr/20

$$\int \frac{{\mathrm{x}}^2+2\mathrm{x}+3}{\sqrt{{\mathrm{x}}^2+\mathrm{x}+1}}\mathrm{dx}$$$$=\int \frac{{\mathrm{x}}^2+\mathrm{x}+1+\mathrm{x}+2}{\sqrt{{\mathrm{x}}^2+\mathrm{x}+1}}\mathrm{dx}$$$$=\int \frac{{\mathrm{x}}^2+\mathrm{x}+1}{\sqrt{{\mathrm{x}}^2+\mathrm{x}+1}}\mathrm{dx}+\int \frac{\mathrm{x}+2}{\sqrt{{\mathrm{x}}^2+\mathrm{x}+1}}\mathrm{dx}$$$$=\int \sqrt{{\mathrm{x}}^2+\mathrm{x}+1}\mathrm{dx}+\int \frac{\frac{1}{2}(2\mathrm{x}+1)+\frac{3}{2}}{\sqrt{{\mathrm{x}}^2+\mathrm{x}+1}}\mathrm{dx}$$$$=\int \sqrt{{\mathrm{x}}^2+2\mathrm{x}.\frac{1}{2}+\frac{1}{4}-\frac{1}{4}+}1\mathrm{dx}+\frac{1}{2}\int \frac{(2\mathrm{x}+1)}{\sqrt{{\mathrm{x}}^2+\mathrm{x}+1}}\mathrm{dx}+\frac{3}{2}\int \frac{1}{\sqrt{{(\mathrm{x}+\frac{1}{2})}^2+\frac{3}{2}}}\mathrm{dx}$$$$=\int \sqrt{{(\mathrm{x}+\frac{1}{2})}^2+\frac{3}{4}}\mathrm{dx}+\frac{1}{2}.2\sqrt{{\mathrm{x}}^2+\mathrm{x}+1}+\mathrm{C}+\frac{3}{2}\int \frac{1}{\sqrt{{(\mathrm{x}+\frac{1}{2})}^2+{\left(\frac{\sqrt{3}}{2}\right)}^2}}\mathrm{dx}$$$$=\int \sqrt{{(\mathrm{x}+\frac{1}{2})}^2+{\left(\frac{\sqrt{3}}{2}\right)}^2}\mathrm{dx}+\frac{3}{2}\int \frac{1}{\sqrt{{(\mathrm{x}+\frac{1}{2})}^2+{\left(\frac{\sqrt{3}}{2}\right)}^2}}\mathrm{dx}+\sqrt{{\mathrm{x}}^2+\mathrm{x}+1}+\mathrm{C}$$$$=\frac{(2\mathrm{x}+1)\sqrt{{\mathrm{x}}^2+\mathrm{x}+1}}{4}+\frac{3}{8}\log (\mathrm{x}+\frac{1}{2}+\sqrt{{\mathrm{x}}^2+\mathrm{x}+1})+\frac{3}{2}\log (\mathrm{x}+\frac{1}{2}+\sqrt{{\mathrm{x}}^2+\mathrm{x}+1})+\sqrt{{\mathrm{x}}^2+\mathrm{x}+1}+\mathrm{C}//.$$

Commented by mathmax by abdo last updated on 08/Apr/20

$$A=\int \frac{x^2+2x+3}{\sqrt{x^2+x+1}}dx\Rightarrow A=\int \frac{x^2+2}{\sqrt{x^2+x+1}}dx+\int \frac{2x+1}{\sqrt{x^2+x+1}}dx$$$$wehave\int \frac{2x+1}{\sqrt{x^2+x+1}}dx=2\sqrt{x^2+x+1}+c_1$$$$\int \frac{x^2+2}{\sqrt{x^2+x+1}}dx=\int \frac{x^2+2}{\sqrt{{(x+\frac{1}{2})}^2+\frac{3}{4}}}dx{=}_{x+\frac{1}{2}=\frac{\sqrt{3}}{2}sh\left(t\right)}$$$$=\int \frac{{(\frac{\sqrt{3}}{2}sh\left(t\right)-\frac{1}{2})}^2+2}{\frac{\sqrt{3}}{2}ch\left(t\right)}\times \frac{\sqrt{3}}{2}ch\left(t\right)dt$$$$=\frac{1}{4}\int ({(\sqrt{3}sh\left(t\right)-1)}^2+8)dt=2t+\frac{1}{4}\int (3{sh}^{2}t-2\sqrt{3}sh\left(t\right)+1)dt$$$$=2t+\frac{3}{8}\int (ch\left(2t\right)-1)dt-\frac{\sqrt{3}}{2}ch\left(t\right)+\frac{t}{4}$$$$=\frac{9}{4}t+\frac{3}{16}sh\left(2t\right)-\frac{3t}{8}-\frac{\sqrt{3}}{3}ch\left(t\right)+C_2$$$$=\frac{15}{8}t+\frac{3}{8}sh\left(t\right)ch\left(t\right)-\frac{\sqrt{3}}{3}ch\left(t\right)+c_2$$$$=\frac{15}{8}argsh\left(\frac{2x+1}{\sqrt{3}}\right)+\frac{3}{8}\left(\frac{2x+1}{\sqrt{3}}\right)\sqrt{1+{\left(\frac{2x+1}{\sqrt{3}}\right)}^2}-\frac{\sqrt{3}}{3}\sqrt{1+{\left(\frac{2x+1}{\sqrt{3}}\right)}^2}+C_2$$$$\Rightarrow A=2\sqrt{x^2+x+1}+\frac{15}{8}ln(\frac{2x+1}{\sqrt{3}}+\sqrt{1+{\left(\frac{2x+1}{\sqrt{3}}\right)}^2}$$$$+\frac{3}{8}\left(\frac{2x+1}{\sqrt{3}}\right)\sqrt{1+{\left(\frac{2x+1}{\sqrt{3}}\right)}^2}-\frac{\sqrt{3}}{3}\sqrt{1+{\left(\frac{2x+1}{\sqrt{3}}\right)}^2}+C$$

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