Tìm: a)nguyên hàm (x - 2) x^2 dx b)ngyên hàm (x - 1)(3x+1) dx c) nguyên hàm can3x^2 dx; d)

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Tìm:

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

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

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

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

Nhập câu trả lời ở đây

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Giải thích

a) \[\int {{{\left( {x - 2} \right)}^2}dx} = \int {\left( {{x^2} - 4x + 4} \right)dx} \]

\[ = \int {{x^2}dx - \int {4xdx + \int {4dx} } } \]

\[ = \frac{{{x^3}}}{3} - 2{x^2} + 4x + C\].

b) \[\int {\left( {x - 1} \right)\left( {3x + 1} \right)dx} = \int {\left( {3{x^2} - 2x - 1} \right)dx} \]

\[ = \int {3{x^2}dx - \int {2xdx - \int {1dx} } } \]

= x3 – x2 + x + C.

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

d) \[\int {\frac{{{{\left( {1 - x} \right)}^2}}}{{\sqrt x }}dx} = \int {\frac{{{x^2} - 2x + 1}}{{\sqrt x }}} dx\]

\[ = \int {\left( {x\sqrt x - 2\sqrt x + \frac{1}{{\sqrt x }}} \right)dx} \]

\[ = \int {\left( {{x^{ - \frac{1}{2}}} + {x^{\frac{1}{2}}} + {x^{\frac{3}{2}}}} \right)dx} \]

\[ = 2{x^{\frac{1}{2}}} - 2.\frac{2}{3}{x^{\frac{3}{2}}} + \frac{2}{5}{x^{\frac{5}{2}}} + C\]

\[ = 2\sqrt x - \frac{4}{3}x\sqrt x + \frac{2}{5}{x^2}\sqrt x + C.\]