Câu hỏi

Tính tích phân:

Tính tích phân: display style integral subscript 0 end subscript superscript 1 end superscript square root of open parentheses 1 minus x to the power of 2 end exponent close parentheses to the power of 3 end exponent end root d x

R. Robo.Ctvx30

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Giải thích

Ta có: (1) Đặt x = sin t ⇒ d x = cos d t Khi x = 0 ⇒ t = 0 x = 1 ⇔ t = 2 π Do đó (1):

Ta có: display style integral subscript 0 end subscript superscript 1 end superscript square root of open parentheses 1 minus x to the power of 2 end exponent close parentheses to the power of 3 end exponent end root d x (1)

Đặt $x = sin t \Rightarrow d x = cos d t$

Khi $x = 0 \Rightarrow t = 0 x = 1 \Leftrightarrow t = \frac{π}{2}$

Do đó (1):

$display style table attributes columnalign left columnspacing 1em rowspacing 4 pt end attributes row cell I equals integral subscript 0 end subscript superscript pi over 2 end superscript square root of open parentheses 1 minus sin to the power of 2 end exponent invisible function application t close parentheses to the power of 3 end exponent end root cos invisible function application t d t end cell row cell equals integral subscript 0 end subscript superscript pi over 2 end superscript cos to the power of 4 end exponent invisible function application t d t equals integral subscript 0 end subscript superscript pi over 2 end superscript open parentheses fraction numerator 1 plus cos invisible function application 2 t over denominator 2 end fraction close parentheses to the power of 2 end exponent d t end cell row cell equals 1 fourth integral subscript 0 end subscript superscript pi over 3 end superscript open parentheses 1 plus 2 cos invisible function application 2 t plus cos to the power of 2 end exponent invisible function application 2 t close parentheses d t end cell row cell equals 1 fourth integral subscript 0 end subscript superscript pi over 2 end superscript open parentheses 1 plus 2 cos invisible function application 2 t plus fraction numerator 1 plus cos invisible function application 4 t over denominator 2 end fraction close parentheses d t end cell end table$