Câu hỏi

Chứng minh rằng I = ∫ − 1 1 ( 4 1 + x 4 + 3 1 + x 3 + 1 + x 2 + 4 1 − x 4 + 3 1 − x 3 + 1 − x 2 ) d x < 12

Chứng minh rằng

$I = \int_{- 1}^{1} (4 1 + x^{4} + 3 1 + x^{3} + 1 + x^{2} + 4 1 - x^{4} + 3 1 - x^{3} + 1 - x^{2}) d x < 12$

R. Robo.Ctvx9

Giáo viên

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Giải thích

$fraction numerator plus open curly brackets table attributes columnalign left end attributes row cell fourth root of 1 minus x to the power of 4 end root equals cube root of open parentheses 1 minus x to the power of 4 close parentheses.1.1.1 end root less or equal than begin display style fraction numerator open parentheses 1 minus x to the power of 4 close parentheses plus 1 plus 1 plus 1 over denominator 4 end fraction end style end cell row cell equals begin display style fraction numerator 4 minus x to the power of 4 over denominator 4 end fraction end style end cell row cell fourth root of 1 plus x to the power of 4 end root equals cube root of open parentheses 1 plus x to the power of 4 close parentheses.1.1.1 end root less or equal than begin display style fraction numerator open parentheses 1 plus x to the power of 4 close parentheses plus 1 plus 1 plus 1 over denominator 4 end fraction end style end cell row cell equals begin display style fraction numerator 4 plus x to the power of 4 over denominator 4 end fraction end style end cell row cell cube root of 1 minus x cubed end root equals cube root of open parentheses 1 minus x cubed close parentheses.1.1 end root less or equal than begin display style fraction numerator open parentheses 1 minus x cubed close parentheses plus 1 plus 1 over denominator 3 end fraction end style end cell row cell equals begin display style fraction numerator 3 minus x cubed over denominator 3 end fraction end style end cell row cell cube root of 1 plus x cubed end root equals cube root of open parentheses 1 plus x cubed close parentheses.1.1 end root less or equal than begin display style fraction numerator open parentheses 1 plus x cubed close parentheses plus 1 plus 1 over denominator 3 end fraction end style end cell row cell equals begin display style fraction numerator 3 plus x cubed over denominator 3 end fraction end style end cell row cell square root of 1 minus x squared end root equals square root of open parentheses 1 minus x squared close parentheses.1 end root less or equal than begin display style fraction numerator open parentheses 1 minus x squared close parentheses plus 1 over denominator 2 end fraction end style equals begin display style fraction numerator 2 minus x squared over denominator 2 end fraction end style end cell row cell square root of 1 plus x squared end root equals square root of open parentheses 1 plus x squared close parentheses.1 end root less or equal than begin display style fraction numerator open parentheses 1 plus x squared close parentheses plus 1 over denominator 2 end fraction end style equals begin display style fraction numerator 2 plus x squared over denominator 2 end fraction end style end cell end table close over denominator not stretchy rightwards double arrow rightwards double arrow fourth root of 1 minus x to the power of 4 end root plus cube root of 1 minus x cubed end root plus square root of 1 minus x squared end root plus fourth root of 1 plus x to the power of 4 end root end fraction plus cube root of 1 plus x cubed end root plus square root of 1 plus x squared end root less or equal than 6 rightwards double arrow I equals integral subscript negative 1 end subscript superscript 1 left parenthesis fourth root of 1 plus x to the power of 4 end root plus cube root of 1 plus x cubed end root plus square root of 1 plus x squared end root plus fourth root of 1 minus x to the power of 4 end root plus cube root of 1 minus x cubed end root plus square root of 1 minus x squared end root right parenthesis d x less than integral subscript negative 1 end subscript superscript 1 6 d x equals 12$

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