Câu 6
Ta có
$\displaystyle \int (x+3).e^{-2x} dx = -\dfrac{1}{2} \displaystyle \int (x+3) d(e^{-2x})$
$= -\dfrac{1}{2} \left( (x+3).e^{-2x} - \displaystyle \int e^{-2x} d(x+3) \right)$
$= -\dfrac{1}{2} (x+3).e^{-2x} + \dfrac{1}{2} e^{-2x}+c$
$= -\dfrac{1}{2} e^{-2x}(x+2) +c$
$= - \dfrac{1}{4} e^{-2x} (2x + 4)+c$
Vậy $m = n=4$. Do đó $m^2 + n^2 = 32$.
Câu 7
Ta tính
$\displaystyle \int \dfrac{\cos x dx}{5\sin x - 9} = \displaystyle \int \dfrac{d(\sin x)}{5\sin x - 9}$
$= \dfrac{1}{5} \ln|5\sin x - 9| + c$
Vậy $a = 1, b = 5$
Suuy ra $2a-b = -3$.
