If = _0^2√3 _2(x^2+4), dx+ _2^4√2x-4, dx, then ^2 is equal to ________.

Question

If $\alpha =\int _{0}^{2\sqrt{3}}{\log }_{2}({x}^{2}+4)\, dx+\int _{2}^{4}\sqrt{2x-4}\, dx$, then ${\alpha }^{2}$ is equal to \_\_\_\_\_\_\_\_.

Answer: 192

Accepted Answer

As we know that

$\begin{gathered} {\int }_{b}^{a}f(x)dx+{\int }_{f(b)}^{f(a)}{f}^{-1}(x)dx=bf(b)-af(a) \ 	herefore \alpha =(2\sqrt{3})(4)-(0)(2) \ =8\sqrt{3} \ \Rightarrow {\alpha }^{2}=192 \end{gathered}$

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