Biết rằng $\Large \int\limits_0^1\dfrac{2x+3}{2-x}\mathrm{d}x=a\mathrm

Biết rằng $\Large \int\limits_0^1\dfrac{2x+3}{2-x}\mathrm{d}x=a\mathrm{ln}2+b$ với $\Large a, b\in \mathbb{Q}$. Chọn khẳng định đúng trong các khẳng định sau:

• A.

  $\Large a < 5$.

• B.

  $\Large b > 4$.

• C.

  $\Large a^2+b^2 > 50$.

• D.

  $\Large a+b < 1$.

Chọn C
Ta có: $\Large \int\limits_0^1\dfrac{2x+3}{2-x}\mathrm{d}x$ $\Large =\int\limits_0^1\left(-2+\dfrac{7}{2-x}\right)\mathrm{d}x$ $\Large =-2x\big|_0^1-7\mathrm{ln}|2-x|\big|_0^1=-2+7\mathrm{ln}2$

Vậy $\Large \int\limits_0^1\dfrac{2x+3}{2-x}\mathrm{d}x=a\mathrm{ln}2+b$ $\Large \Rightarrow a=7; b=-2$

Khi đó $\Large a^2+b^2=7^2+(-2)^2=53 > 50$.