Mathematics - Definite Integration Question with Solution | TestHub

$\underset{0}{\overset{\frac{1}{2}}{\int }}\frac{\left(1+\sqrt{3}\right)dx}{{\left[{\left(1+x\right)}^2{\left(1-x\right)}^6\right]}^{\frac{1}{4}}}$
$\underset{0}{\overset{\frac{1}{2}}{\int }}\frac{\left(1+\sqrt{3}\right)dx}{{\left(1+x\right)}^2{\left[\frac{{\left(1-x\right)}^6}{{\left(1+x\right)}^6}\right]}^{\frac{1}{4}}}$
Put$\frac{1-x}{1+x}=t\Rightarrow \frac{-2dx}{{\left(1+x\right)}^2}=dt$
$I=\underset{1}{\overset{\frac{1}{3}}{\int }}\frac{\left(1+\sqrt{3}\right)dt}{-2t^{\frac{6}{4}}}=\frac{-\left(1+\sqrt{3}\right)}{2}\times {\left|-\frac{2}{\sqrt{t}}\right|}_1^{\frac{1}{3}}=\left(1+\sqrt{3} \right)\left(\sqrt{3}-1\right)=2$