# Integrate: 1 / (1+ square root x) ^8

Solution:

$I=\int \frac{1}{(1+\sqrt{x})^{8}}dx \hspace{0.5cm}put \hspace{0.5cm}1+\sqrt{x}=t \\ \\ \Rightarrow \hspace{0.5cm}\frac{1}{2\sqrt{x}}dx=dt\ \\ \Rightarrow \hspace{0.5cm}I=\int \frac{2\sqrt{x}}{t^{8}}dt\ \\ \Rightarrow \hspace{0.5cm}I=2\int \frac{t-1}{t^{8}}dt\ \\ \Rightarrow \hspace{0.5cm}I=2\int (\frac{1}{t^{7}}-\frac{1}{t^{8}})dt$

$\ \\ \Rightarrow \hspace{0.5cm}I=2[-\frac{1}{6t^{6}}-\frac{1}{7t^{7}}]+c\ \\ \Rightarrow \hspace{0.5cm}I=2[-\frac{1}{6x^{3}}-\frac{1}{7x^\frac{7}{2}}]+c$

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