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

Question

If $\alpha = 2\sqrt3\displaystyle\int_0^{2\sqrt3}\log_2(x^2+4)\,dx + \int_2^4\sqrt{2x-4}\,dx$, then $\alpha^2$ is equal to \_\_\_\_\_\_\_\_.

Answer: 192

Accepted Answer

Using the standard property for inverse-function pairs, $\displaystyle\int_a^b f(x)\,dx + \int_{f(a)}^{f(b)} f^{-1}(x)\,dx = bf(b) - af(a)$, with $f(x)=\log_2(x^2+4)$, $a=0,\,b=2\sqrt3$ (so $f(0)=2$, $f(2\sqrt3)=4$):

$$\alpha = (2\sqrt3)(4) - (0)(2) = 8\sqrt3$$

$$\alpha^2 = 192$$

Solutions · 1