Đáp án:

$\dfrac{\sqrt{(x^2+5)^3} - 9\sqrt{x^2+5} + 3C}{\sqrt{(x^2+5)^3} - 9\sqrt{x^2+5} + 3C_3}$

Giải thích các bước giải:

$\bullet\quad I_1 = \displaystyle\int x(x^2+5)^{\tfrac12}dx$

$\to I_1 = \displaystyle\int x\sqrt{x^2+5}dx$

$\to I_1= \displaystyle\int \dfrac{2x (x^2+5)}{2\sqrt{x^2+5}}dx$

$\to I_1 = \displaystyle\int \sqrt{(x^2+5)^2}d\left(\sqrt{x^2+5}\right)$

$\to I_1 = \dfrac{\sqrt{(x^2+5)^3}}{3} + C_1$

$\bullet\quad I_2 = 3\displaystyle\int x(x^2+5)^{-\tfrac12}dx$

$\to I_2 = 3\displaystyle\int\dfrac{2x}{2\sqrt{x^2+5}}dx$

$\to I_2 = 3\displaystyle\int d\left(\sqrt{x^2+5}\right)dx$

$\to I_2 = 3\sqrt{x^2+5} + C_2$

$\bullet\quad I_3 = \displaystyle\int \dfrac{x[(x^2+5) - 3]}{\sqrt{x^2+5}}dx$

$\to I_3 = \displaystyle\int\left[\dfrac{2x(x^2+5)}{2\sqrt{x^2+5}} - 3\cdot\dfrac{2x}{2\sqrt{x^2+5}}\right]dx$

$\to I_3 = \dfrac{\sqrt{(x^2+5)^3}}{3} - 3\sqrt{x^2 + 5} + C_3$

Ta được:

$\quad I =\dfrac{I_1-I_2}{I_3}$

$\to I = \dfrac{\dfrac{\sqrt{(x^2+5)^3}}{3} - 3\sqrt{x^2+5} + C}{\dfrac{\sqrt{(x^2+5)^3}}{3} - 3\sqrt{x^2 + 5} + C_3}$

$\to I = \dfrac{\sqrt{(x^2+5)^3} - 9\sqrt{x^2+5} + 3C}{\sqrt{(x^2+5)^3} - 9\sqrt{x^2+5} + 3C_3}$