Câu hỏi

Tính tích phân

Tính tích phân $display style integral subscript pi end subscript superscript fraction numerator 4 pi over denominator 3 end fraction end superscript fraction numerator d x over denominator sin invisible function application x over 2 end fraction$

R. Robo.Ctvx30

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Giải thích

I = ∫ π 3 4 π + 3 m u sin 2 x d x = 2 ∫ π 3 4 π + 3 m u 1 − cos 2 2 x sin 2 x ⋅ d 2 x = − 2 ∫ π 3 4 π + 3 m u 1 − cos 2 2 x d cos 2 x

$I = \int_{π}^{\frac{4 π}{3}} + 3 m u \frac{d x}{sin \frac{x}{2}} = 2 \int_{π}^{\frac{4 π}{3}} + 3 m u \frac{sin \frac{x}{2} \cdot d \frac{x}{2}}{1 - cos ^{2} \frac{x}{2}} = - 2 \int_{π}^{\frac{4 π}{3}} + 3 m u \frac{d cos \frac{x}{2}}{1 - cos ^{2} \frac{x}{2}}$

$table attributes columnalign left columnspacing 1em end attributes row cell text Đặt end text t equals cos invisible function application x over 2 not stretchy rightwards double arrow I equals negative 2 integral subscript 0 superscript negative 1 divided by 2 end superscript fraction numerator d t over denominator 1 minus t squared end fraction end cell row cell equals integral subscript negative 1 half end subscript superscript 0 open parentheses fraction numerator 1 over denominator 1 minus t end fraction plus fraction numerator 1 over denominator 1 plus t end fraction close parentheses d t end cell row cell I equals open ln invisible function application fraction numerator 1 plus t over denominator 1 minus t end fraction close vertical bar subscript negative 1 divided by 2 end subscript superscript 0 equals ln invisible function application 3. end cell end table$