Đáp án:
$P = 46$
Giải thích các bước giải:
$\quad I =\displaystyle\int\limits_1^2\dfrac{dx}{(x+1)\sqrt x + x\sqrt{x+1}}$
$\to I =\displaystyle\int\limits_1^2\dfrac{1}{\sqrt{x(x+1)}(\sqrt{x+1} + \sqrt x)}dx$
$\to I =\displaystyle\int\limits_1^2\dfrac{\sqrt{x+1} -\sqrt x}{\sqrt{x(x+1)}}dx$
$\to I =\displaystyle\int\limits_1^2\left(\dfrac{1}{\sqrt x} -\dfrac{1}{\sqrt{x+1}}\right)dx$
$\to I =\displaystyle\int\limits_1^2\dfrac{1}{\sqrt x}dx - \displaystyle\int\limits_1^2\dfrac{1}{\sqrt{x+1}}dx$
$\to I = 2\sqrt x\Bigg|_1^2 - 2\sqrt{x+1}\Bigg|_1^2$
$\to I = 2(\sqrt2 - 1) - 2(\sqrt3 - \sqrt2)$
$\to I = 4\sqrt2 - 2\sqrt3 - 2$
$\to I = \sqrt{32} - \sqrt{12} - 2$
$\to \begin{cases}a = 32\\b = 12\\c = 2\end{cases}$
$\to P = a + b + c = 32 + 12 + 2 = 46$
