Giải bài 63 trang 30 sách bài tập toán 12 - Cánh diều

Đề bài

Tính:

a) \(\int\limits_0^1 { - 2dx} \);

b) \(\int\limits_0^1 {\frac{{2x}}{3}dx} \);

c) \(\int\limits_0^1 {{x^4}dx} \);

d) \(\int\limits_1^3 {2\sqrt[3]{x}dx} \);

e) \(\int\limits_1^2 {\frac{2}{{3x}}dx} \);

g) \(\int\limits_1^9 {\left( {x\sqrt x  - 2} \right)dx} \).

Lời giải chi tiết

a) \(\int\limits_0^1 { - 2dx}  = \left. { - 2{\rm{x}}} \right|_0^1 =  - 2\).

b) \(\int\limits_0^1 {\frac{{2x}}{3}dx}  = \left. {\frac{2}{3}{\rm{.}}\frac{{{{\rm{x}}^2}}}{2}} \right|_0^1 = \left. {\frac{{{{\rm{x}}^2}}}{3}} \right|_0^1 = \frac{1}{3}\).

c) \(\int\limits_0^1 {{x^4}dx}  = \left. {\frac{{{x^5}}}{5}} \right|_0^1 = \frac{1}{5}\).

d) \(\int\limits_1^3 {2\sqrt[3]{x}dx}  = \int\limits_1^3 {2{x^{\frac{1}{3}}}dx}  = \left. {\frac{{2{{\rm{x}}^{\frac{4}{3}}}}}{{\frac{4}{3}}}} \right|_1^3 = \left. {\frac{{3x\sqrt[3]{x}}}{2}} \right|_1^3 = \frac{{9\sqrt[3]{3} - 3}}{2}\).

e) \(\int\limits_1^2 {\frac{2}{{3x}}dx}  = \left. {\frac{2}{3}\ln \left| x \right|} \right|_1^2 = \frac{2}{3}\ln 2\).

g) \(\int\limits_1^9 {\left( {x\sqrt x  - 2} \right)dx}  = \int\limits_1^9 {\left( {{x^{\frac{3}{2}}} - 2} \right)dx}  = \left. {\left( {\frac{{{{\rm{x}}^{\frac{5}{2}}}}}{{\frac{5}{2}}} - 2{\rm{x}}} \right)} \right|_1^9 = \left. {\left( {\frac{{2{{\rm{x}}^2}\sqrt x }}{5} - 2{\rm{x}}} \right)} \right|_1^9 = \frac{{396}}{5} - \left( { - \frac{8}{5}} \right) = \frac{{404}}{5}\).