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Question Number 96317 by  M±th+et+s last updated on 31/May/20

$$findthesum$$$$\frac{1}{4}+\frac{1\times 3}{4\times 6}+\frac{1\times 3\times 5}{4\times 6\times 8}.....=?$$$$find$$$$\int \frac{\sqrt{x}}{\sqrt{x}+\sqrt{3-x}}dx$$$$$$$$$$

Commented by bemath last updated on 01/Jun/20

$$\left(2\right)\int \frac{\sqrt{\mathrm{x}}\mathrm{dx}}{\sqrt{\mathrm{x}}(1+\sqrt{\frac{3}{\mathrm{x}}-1})}=$$$$\int \frac{\mathrm{dx}}{1+\sqrt{\frac{3}{\mathrm{x}}-1}};[\frac{3}{\mathrm{x}}-1={\mathrm{z}}^2]$$$$\mathrm{x}=\frac{3}{{\mathrm{z}}^2+1}\Rightarrow \mathrm{dx}=\frac{-6\mathrm{z}}{{({\mathrm{z}}^2+1)}^2}\mathrm{dz}$$$$\int \frac{-6\mathrm{z}\mathrm{dz}}{{({\mathrm{z}}^2+1)}^2(1+\mathrm{z})}=-\frac{3}{2}\int (\frac{1}{\mathrm{z}+1}-\frac{\mathrm{z}+1}{{\mathrm{z}}^2+1}-\frac{2\mathrm{z}-2}{{({\mathrm{z}}^2+1)}^2})\mathrm{dz}$$$$=-\frac{3}{4}[2\ln \mid \sqrt{\frac{3-\mathrm{x}}{\mathrm{x}}}\mid +\ln \mid \frac{\mathrm{x}}{3}\mid +\frac{2\mathrm{x}}{3}(\sqrt{\frac{3-\mathrm{x}}{\mathrm{x}}}+2)]+\mathrm{c}$$

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