$2)\quad I = \displaystyle\int\limits_2^4dx\displaystyle\int\limits_x^{2x}\dfrac{y}{x}dy$
$\Leftrightarrow I = \displaystyle\int\limits_2^4\left(\dfrac{y^2}{2x}\Bigg|_x^{2x}\right)dx$
$\Leftrightarrow I = \displaystyle\int\limits_2^4\dfrac{3x}{2}dx$
$\Leftrightarrow I = \dfrac{3x^2}{4}\Bigg|_2^4$
$\Leftrightarrow I = 9$
$3)\quad I = \displaystyle\iint\limits_D\dfrac{\ln y}{x+1}dxdy\quad \text{với}\quad D:\begin{cases}x = 0\\x = 1\\y = 1\\y = e\end{cases}$
Miền $D$ được biểu diễn:
$D = \{(x,y): 0 \leqslant x \leqslant 1;\ 1 \leqslant y \leqslant e\}$
Ta được:
$\quad I = \displaystyle\int\limits_0^1dx\displaystyle\int\limits_1^e\dfrac{\ln y}{x+1}dy$
$\Leftrightarrow I = \displaystyle\int\limits_0^1\left[\dfrac{y}{x+1}(\ln y +1)\Bigg|_1^e\right]dx$
$\Leftrightarrow I = \displaystyle\int\limits_0^1\dfrac{2e-1}{x+1}dx$
$\Leftrightarrow I = (2e-1)\ln(x+1)\Bigg|_0^1$
$\Leftrightarrow I = (2e-1)\ln 2$
$4)\quad I = \displaystyle\iint\limits_D\dfrac{dxdy}{(x+y)^2}$
với $D:\{(x,y): 3 \leqslant x \leqslant 5;\ 1 \leqslant y \leqslant 2\}$
Ta được:
$\quad I = \displaystyle\int\limits_3^5dx\displaystyle\int\limits_1^2\dfrac{dy}{(x+y)^2}$
$\Leftrightarrow I = \displaystyle\int\limits_3^5\left(-\dfrac{1}{x+y}\Bigg|_1^2\right)dx$
$\Leftrightarrow I = \displaystyle\int\limits_3^5\left(\dfrac{1}{x+1} -\dfrac{1}{x+2}\right)dx$
$\Leftrightarrow I = \ln\dfrac{x+1}{x+2}\Bigg|_3^5$
$\Leftrightarrow I = \ln\dfrac{15}{14}$
$5)\quad I = \displaystyle\iint\limits_D12ydxdy\quad \text{với}\quad D:\begin{cases}x= y^2\\x = y\end{cases}$
Phương trình tung độ giao điểm:
$\quad y^2 = y \Leftrightarrow \left[\begin{array}{l}y = 0\\y = 1\end{array}\right.$
Miền $D$ được biểu diễn:
$D:\{(x,y): 0 \leqslant y \leqslant 1;\ y^2 \leqslant x \leqslant y\}$
Ta được:
$\quad I = \displaystyle\int\limits_0^1dy\displaystyle\int\limits_{y^2}^y12ydx$
$\Leftrightarrow I = \displaystyle\int\limits_0^1\left(12xy\Bigg|_{y^2}^y\right)dy$
$\Leftrightarrow I = \displaystyle\int\limits_0^1(12y^2 - 12y^3)dy$
$\Leftrightarrow I = (4y^3 - 3y^4)\Bigg|_0^1$
$\Leftrightarrow I = 1$
