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Evaluate\[\int \frac{(x+2) d x}{\sqrt{x^{2}+2 x+3}}\] |
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Answer» \(\int\frac{(x+2)dx}{\sqrt{x^2+2x+3}}\) = \(\frac12\int\frac{(2x+4)dx}{\sqrt{x^2+2x+3}}\) = \(\frac12\left(\int\frac{2x+2}{\sqrt{x^2+2x+3}}dx+\int\frac{2}{\sqrt{x^2+2x+3}}dx\right)\) = \(\frac12(2\sqrt{x^2+2x+3}+\int\frac{2}{(x+1)^2+2}dx)\) = \(\sqrt{x^2+2x+3}+log|(x+1)+\sqrt{x^2+2x+3}|+c\) |
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