Giải bài 6 trang 15 sách bài tập toán 12 - Chân trời sáng tạo

Đề bài

Tính:

a) $A = \int\limits_{ - 1}^2 {\left( {x - 4{{\rm{x}}^2}} \right)dx}  + 4\int\limits_{ - 1}^2 {\left( {{x^2} - 1} \right)dx}$;

b) $B = \int\limits_{ - 1}^0 {\left( {{x^3} - 6{\rm{x}}} \right)dx}  + \int\limits_0^1 {\left( {{t^3} - 6{\rm{t}}} \right)dt}$.

Lời giải chi tiết

a)

$\begin{array}{l}A = \int\limits_{ - 1}^2 {\left( {x - 4{{\rm{x}}^2}} \right)dx}  + 4\int\limits_{ - 1}^2 {\left( {{x^2} - 1} \right)dx}  = \int\limits_{ - 1}^2 {\left( {x - 4{{\rm{x}}^2}} \right)dx}  + \int\limits_{ - 1}^2 {\left( {4{x^2} - 4} \right)dx}  = \int\limits_{ - 1}^2 {\left( {x - 4{{\rm{x}}^2} + 4{x^2} - 4} \right)dx} \\ = \int\limits_{ - 1}^2 {\left( {x - 4} \right)dx}  = \left. {\left( {\frac{{{x^2}}}{2} - 4x} \right)} \right|_{ - 1}^2 = \left( {\frac{{{2^2}}}{2} - 4.2} \right) - \left( {\frac{{{{\left( { - 1} \right)}^2}}}{2} - 4.\left( { - 1} \right)} \right) =  - \frac{{21}}{2}\end{array}$

b)

$\begin{array}{l}B = \int\limits_{ - 1}^0 {\left( {{x^3} - 6{\rm{x}}} \right)dx}  + \int\limits_0^1 {\left( {{t^3} - 6{\rm{t}}} \right)dt}  = \int\limits_{ - 1}^0 {\left( {{x^3} - 6{\rm{x}}} \right)dx}  + \int\limits_0^1 {\left( {{x^3} - 6{\rm{x}}} \right)dx} \\ = \int\limits_{ - 1}^1 {\left( {{x^3} - 6{\rm{x}}} \right)dx}  = \left. {\left( {\frac{{{x^4}}}{4} - 3{{\rm{x}}^2}} \right)} \right|_{ - 1}^1 = \left( {\frac{{{1^4}}}{4} - {{3.1}^2}} \right) - \left( {\frac{{{{\left( { - 1} \right)}^4}}}{4} - 3.{{\left( { - 1} \right)}^2}} \right) = 0\end{array}$