The value of the integral ∫01/2 1+3(x+1)2(1−x)61/4dx is__.

# The value of the integral ${\int }_{0}^{1/2} \frac{1+\sqrt{3}}{{\left(\left(x+1{\right)}^{2}\left(1-x{\right)}^{6}\right)}^{1/4}}dx$ is__.

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### Solution:

$\begin{array}{c}I={\int }_{0}^{1/2} \frac{1+\sqrt{3}}{{\left(\left(x+1{\right)}^{2}\left(1-x{\right)}^{6}\right)}^{1/4}}dx\\ ={\int }_{0}^{1/2} \frac{\left(1+\sqrt{3}\right)dx}{\left(1+x{\right)}^{2}{\left[\frac{\left(1-x{\right)}^{6}}{\left(1+x{\right)}^{6}}\right]}^{1/4}}\end{array}$

Put  $\frac{1-x}{1+x}=t$

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