Giải bài 1 trang 8 sách bài tập toán 12 - Chân trời sáng tạo

Đề bài

Tìm:

a) \(\int {{{\left( {x - 2} \right)}^2}dx} \);

b) \(\int {\left( {x - 1} \right)\left( {3{\rm{x}} + 1} \right)dx} \);

c) \(\int {\sqrt[3]{{{x^2}}}dx} \);

d) \(\int {\frac{{{{\left( {1 - x} \right)}^2}}}{{\sqrt x }}dx} \).

Lời giải chi tiết

a) \(\int {{{\left( {x - 2} \right)}^2}dx}  = \int {\left( {{x^2} - 4{\rm{x}} + 4} \right)dx}  = \frac{{{x^3}}}{3} - 2{{\rm{x}}^2} + 4{\rm{x}} + C\).

b) \(\int {\left( {x - 1} \right)\left( {3{\rm{x}} + 1} \right)dx}  = \int {\left( {3{{\rm{x}}^2} - 2{\rm{x}} - 1} \right)dx}  = {x^3} - {x^2} - x + C\).

c) \(\int {\sqrt[3]{{{x^2}}}dx}  = \int {{x^{\frac{2}{3}}}dx}  = \frac{{{x^{\frac{5}{3}}}}}{{\frac{5}{3}}} + C = \frac{3}{5}{x^{\frac{5}{3}}} + C = \frac{3}{5}x.\sqrt[3]{{{x^2}}} + C\).

d) \(\begin{array}{l}\int {\frac{{{{\left( {1 - x} \right)}^2}}}{{\sqrt x }}dx}  = \int {\frac{{{x^2} - 2{\rm{x}} + 1}}{{{x^{\frac{1}{2}}}}}dx}  = \int {\left( {{x^{\frac{3}{2}}} - 2{{\rm{x}}^{\frac{1}{2}}} + {{\rm{x}}^{ - \frac{1}{2}}}} \right)dx}  = \frac{{{x^{\frac{5}{2}}}}}{{\frac{5}{2}}} - \frac{{2{{\rm{x}}^{\frac{3}{2}}}}}{{\frac{3}{2}}} + \frac{{{{\rm{x}}^{\frac{1}{2}}}}}{{\frac{1}{2}}} + C\\ = \frac{2}{5}{x^{\frac{5}{2}}} - \frac{4}{3}{{\rm{x}}^{\frac{3}{2}}} + 2{{\rm{x}}^{\frac{1}{2}}} + C = \frac{2}{5}{x^2}\sqrt x  - \frac{4}{3}{\rm{x}}\sqrt x  + 2\sqrt x  + C\end{array}\)