Cho hàm số $\Large f(x)=\left\{\begin{align} & x^2+2x\ \text{khi}\ x\l

Cho hàm số $\Large f(x)=\left\{\begin{align} & x^2+2x\ \text{khi}\ x\leq 1\\ & 2x+1\ \text{khi}\ x > 1\end{align}\right.$. Tích phân $\Large I=\int\limits_{e^{-2}}^e\dfrac{f(\ln x+2)}{x}dx$ bằng

• A.

  $\Large 18$.

• B.

  $\Large \dfrac{34}{3}$.

• C.

  $\Large 12$.

• D.

  $\Large \dfrac{56}{3}$.

Chọn B

Đặt $\Large t=\ln x+2\Rightarrow dt=\dfrac{dx}{x}$.

Đổi cận: $\Large \left\{\begin{align} & x=e^{-2}\Rightarrow t=0\\ & x=e\Rightarrow t=3\end{align}\right.$.

Khi đó

$\Large I=\int\limits_0^3f(t)dt=\int\limits_0^3f(x)dx$

$\large  = \int\limits_0^1 {f\left( x \right)dx + } \int\limits_1^3 {f\left( x \right)dx}$.

Ta có $\Large I_1=\int\limits_0^1f(x)dx=\int\limits_0^1(x^2+2x)dx=\dfrac{4}{3}$.

Ta có $\Large I_2=\int\limits_1^3f(x)dx=\int\limits_1^3(2x+1)dx=10$.

Vậy $\Large I=I_1+I_2=\dfrac{4}{3}+10=\dfrac{34}{3}$.