Cho các mệnh đề sau:

9/25

1. \(\int {\left( {\sqrt[3]{{{x^2}}} + x - 2} \right){\rm{d}}x} = \frac{3}{5}\sqrt[3]{{{x^5}}} + \frac{1}{2}{x^2} - 2x + C\).

ĐúngSai

2. \(\int {\frac{1}{{2023{x^{2024}}}}{\rm{d}}x} = \frac{1}{{{{2023}^2}{x^{2023}}}} + C\).

ĐúngSai

3. \(\int {{{\left( {2x - 2024} \right)}^2}{\rm{d}}x} = x - 1012 + C\).

ĐúngSai

4. \(\int {\left( {\frac{1}{4}{x^4} + 4{x^3}} \right){\rm{d}}x} = \frac{1}{{20}}{x^5} + \frac{4}{3}{x^4} + C\).

ĐúngSai

Giải thích

1. Đúng. \(\int {\left( {\sqrt[3]{{{x^2}}} + x - 2} \right){\rm{d}}x} = \int {\left( {{x^{\frac{2}{3}}} + x - 2} \right){\rm{d}}x} = \frac{3}{5}{x^{\frac{5}{3}}} + \frac{1}{2}{x^2} - 2x + C = \frac{3}{5}\sqrt[3]{{{x^5}}} + \frac{1}{2}{x^2} - 2x + C\).

2. Sai. \(\int {\frac{1}{{2023{x^{2024}}}}{\rm{d}}x} = \frac{1}{{2023}}\int {{x^{ - 2024}}{\rm{d}}x} = \frac{{ - 1}}{{{{2023}^2}{x^{2023}}}} + C\).

3. Sai. \(\int {{{\left( {2x - 2024} \right)}^2}{\rm{d}}x} = \frac{1}{6}{\left( {2x - 2024} \right)^3} + C\).

4. Sai. \(\int {\left( {\frac{1}{4}{x^4} + 4{x^3}} \right){\rm{d}}x} = \frac{1}{{20}}{x^5} + {x^4} + C\).