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Question Number 185848 by Mingma last updated on 28/Jan/23

Answered by Frix last updated on 28/Jan/23

$$\int \sqrt{{\mathrm{e}}^x+2}dx\stackrel{t=\sqrt{{\mathrm{e}}^x+2}}{=}2\int \frac{t^2}{t^2-2}dt=$$$$=2\int dt+\sqrt{2}\int \frac{dt}{t-\sqrt{2}}-\sqrt{2}\int \frac{dt}{t+\sqrt{2}}=$$$$=2t+\sqrt{2}\ln (t-\sqrt{2})-\sqrt{2}\ln (t+\sqrt{2})=$$$$...$$$$=2\sqrt{{\mathrm{e}}^x+2}-\sqrt{2}x+2\sqrt{2}\ln (\sqrt{{\mathrm{e}}^x+2}-\sqrt{2})+C$$$$$$

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