Помогите решить задачу по интеграллам пож.)
S dx / 4-5sinx

Автор: chips. Предмет: Алгебра. Опубликовано: 04.07.2019 chips
  1. Udun Пользователь Udun оставил ответ:
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    Универсальная тригонометрическая подстановка.
    <img src=" https://tex.z-dn.net/?f=%5Cdisplaystyle+%5Cint%5Climits+%7B+%5Cfrac%7Bdx%7D%7B4-5%5Csin+x%7D+%7D+%5C%2C+%3D%5Cbigg%5C%7B+tg+%5Cfrac%7Bx%7D%7B2%7D%3Dt%3B%5C%2C%5C%2C+%5Csin+x%3D+%5Cfrac%7B2t%7D%7Bt%5E2%2B1%7D+%3B%5Ccos+x%3D+%5Cfrac%7B1-t%5E2%7D%7B1%2Bt%5E2%7D%3B%5C%5C+%5C%5C+%5C%5C+dx%3D+%5Cfrac%7B2dt%7D%7Bt%5E2%2B1%7D++%5Cbigg%5C%7D%3D%5Cint%5Climits+%7B++%5Cfrac%7Bdt%7D%7B2t%5E2-5t%2B2%7D+%7D+%5C%2C+%3D%5Cint%5Climits+%7B+%5Cfrac%7Bdt%7D%7B%28+%5Csqrt%7B2%7Dt-+%5Cfrac%7B5%7D%7B2+%5Csqrt%7B2%7D+%7D++%29%5E2-+%5Cfrac%7B9%7D%7B8%7D+%7D+%7D+%5C%2C+%3D%5C%5C+%5C%5C+%5C%5C+%3D%5Cbigg%5C%7B+%5Csqrt%7B2%7Dt-+%5Cfrac%7B5%7D%7B2+%5Csqrt%7B2%7D+%7D+%3Du%5Cbigg%5C%7D%3D+%5Cfrac%7B1%7D%7B%5Csqrt%7B2%7D%7D+%5Cint%5Climits+%7B+%5Cfrac%7Bdu%7D%7Bu%5E2-+%5Cfrac%7B9%7D%7B8%7D+%7D+%7D+%5C%2C+%3D+%5Cfrac%7B1%7D%7B%5Csqrt%7B2%7D%7D+%5Ccdot+%5Cfrac%7B%5Csqrt%7B2%7D%7D%7B3%7D+%5Cln%7C+%5Cfrac%7B4u-3+%5Csqrt%7B2%7D+%7D%7B4u%2B3%5Csqrt%7B2%7D%7D%7C%2BC%3D&quot; id=" TexFormula1" onerror=" texError(this)" title=" /displaystyle /int/limits { /frac{dx}{4-5/sin x} } /, =/bigg/{ tg /frac{x}{2}=t;/,/, /sin x= /frac{2t}{t^2+1} ;/cos x= /frac{1-t^2}{1+t^2};// // // dx= /frac{2dt}{t^2+1} /bigg/}=/int/limits { /frac{dt}{2t^2-5t+2} } /, =/int/limits { /frac{dt}{( /sqrt{2}t- /frac{5}{2 /sqrt{2} } )^2- /frac{9}{8} } } /, =// // // =/bigg/{ /sqrt{2}t- /frac{5}{2 /sqrt{2} } =u/bigg/}= /frac{1}{/sqrt{2}} /int/limits { /frac{du}{u^2- /frac{9}{8} } } /, = /frac{1}{/sqrt{2}} /cdot /frac{/sqrt{2}}{3} /ln| /frac{4u-3 /sqrt{2} }{4u+3/sqrt{2}}|+C=" alt=" /displaystyle /int/limits { /frac{dx}{4-5/sin x} } /, =/bigg/{ tg /frac{x}{2}=t;/,/, /sin x= /frac{2t}{t^2+1} ;/cos x= /frac{1-t^2}{1+t^2};// // // dx= /frac{2dt}{t^2+1} /bigg/}=/int/limits { /frac{dt}{2t^2-5t+2} } /, =/int/limits { /frac{dt}{( /sqrt{2}t- /frac{5}{2 /sqrt{2} } )^2- /frac{9}{8} } } /, =// // // =/bigg/{ /sqrt{2}t- /frac{5}{2 /sqrt{2} } =u/bigg/}= /frac{1}{/sqrt{2}} /int/limits { /frac{du}{u^2- /frac{9}{8} } } /, = /frac{1}{/sqrt{2}} /cdot /frac{/sqrt{2}}{3} /ln| /frac{4u-3 /sqrt{2} }{4u+3/sqrt{2}}|+C=" align=" absmiddle" class=" latex-formula" >

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