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Question Number 12908 by b.e.h.i.8.3.4.1.7@gmail.com last updated on 06/May/17

Answered by 433 last updated on 07/May/17

$${\int }_{\sqrt{2}}^2\left(\frac{x^2}{x+[x+1]}\right)dx+{\int }_2^{\sqrt{5}}\left(\frac{x^2}{x+[x+1]}\right)dx=$$$${\int }_{\sqrt{2}}^2\left(\frac{x^2}{x+2}\right)dx+{\int }_2^{\sqrt{5}}\left(\frac{x^2}{x+3}\right)dx=$$$${\int }_{\sqrt{2}}^2(\frac{x^2-4}{x+2}+\frac{4}{x+2})dx+{\int }_2^{\sqrt{5}}(\frac{x^2-9}{x+3}+\frac{9}{x+3})dx=$$$${\int }_{\sqrt{2}}^2(x-2)dx+{\int }_{\sqrt{2}}^2\left(\frac{4}{x+2}\right)dx+{\int }_2^{\sqrt{5}}(x-3)dx+{\int }_2^{\sqrt{5}}\left(\frac{9}{x+3}\right)dx$$$${[\frac{x^2}{2}-2x]}_{\sqrt{2}}^2+{[4\ln \mid x+2\mid ]}_{\sqrt{2}}^2+{[\frac{x^2}{2}-3x]}_2^{\sqrt{5}}+{[9\ln \mid x+3\mid ]}_2^{\sqrt{5}}$$$$(2-4-1+2\sqrt{2}+4\ln 4-4\ln (\sqrt{2}+2))+(\frac{5}{2}-3\sqrt{5}-2+6)+(9\ln (\sqrt{5}+3)-9\ln \left(5\right))$$$$-3+2\sqrt{2}+4\ln \left(\frac{4}{\sqrt{2}+2}\right)+\frac{13}{2}-3\sqrt{5}+9\ln \left(\frac{\sqrt{5}+3}{5}\right)$$

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