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Задание: Найти все первообразные функции!
Вложение!
Огромное спасибо всем кто поможет!
Нужно с 1 по 8
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![](https://st.uroker.com/files/dae/dae80f007c56a9b082de5bbe08d815c4.jpg)
Ответы
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1) ![F(x)= intlimits {[e^{3x+2}-cos(3x)+ frac{1}{cos^2(3x)}] } , dx =
F(x)= intlimits {[e^{3x+2}-cos(3x)+ frac{1}{cos^2(3x)}] } , dx =](https://tex.z-dn.net/?f=F%28x%29%3D+intlimits+%7B%5Be%5E%7B3x%2B2%7D-cos%283x%29%2B+frac%7B1%7D%7Bcos%5E2%283x%29%7D%5D+%7D+%2C+dx+%3D%0A)
![= frac{1}{3} intlimits {[e^{3x+2}-cos(3x)+ frac{1}{cos^2(3x)}] } , d(3x) = [t=3x]= = frac{1}{3} intlimits {[e^{3x+2}-cos(3x)+ frac{1}{cos^2(3x)}] } , d(3x) = [t=3x]=](https://tex.z-dn.net/?f=%3D+frac%7B1%7D%7B3%7D++intlimits+%7B%5Be%5E%7B3x%2B2%7D-cos%283x%29%2B+frac%7B1%7D%7Bcos%5E2%283x%29%7D%5D+%7D+%2C+d%283x%29+%3D+%5Bt%3D3x%5D%3D)
![= frac{1}{3} intlimits {[e^{t+2}-cos(t)+ frac{1}{cos^2(t)}] } , dt = = frac{1}{3} intlimits {[e^{t+2}-cos(t)+ frac{1}{cos^2(t)}] } , dt =](https://tex.z-dn.net/?f=%3D+frac%7B1%7D%7B3%7D++intlimits+%7B%5Be%5E%7Bt%2B2%7D-cos%28t%29%2B+frac%7B1%7D%7Bcos%5E2%28t%29%7D%5D+%7D+%2C+dt+%3D)
![= frac{1}{3}[ intlimits {e^{t+2}}dt- intlimits {cos(t)} , dt+ intlimits {frac{1}{cos^2(t)} , dt ] = = frac{1}{3}[ intlimits {e^{t+2}}dt- intlimits {cos(t)} , dt+ intlimits {frac{1}{cos^2(t)} , dt ] =](https://tex.z-dn.net/?f=%3D+frac%7B1%7D%7B3%7D%5B++intlimits+%7Be%5E%7Bt%2B2%7D%7Ddt-+intlimits+%7Bcos%28t%29%7D+%2C+dt%2B+intlimits+%7Bfrac%7B1%7D%7Bcos%5E2%28t%29%7D+%2C+dt++%5D+%3D)
![= frac{1}{3}[ intlimits {e^{t+2}}d(t+2)- intlimits {cos(t)} , dt+ intlimits {frac{1}{cos^2(t)} , dt ] = = frac{1}{3}[ intlimits {e^{t+2}}d(t+2)- intlimits {cos(t)} , dt+ intlimits {frac{1}{cos^2(t)} , dt ] =](https://tex.z-dn.net/?f=%3D+frac%7B1%7D%7B3%7D%5B++intlimits+%7Be%5E%7Bt%2B2%7D%7Dd%28t%2B2%29-+intlimits+%7Bcos%28t%29%7D+%2C+dt%2B+intlimits+%7Bfrac%7B1%7D%7Bcos%5E2%28t%29%7D+%2C+dt++%5D+%3D)
![=frac{1}{3}[e^{t+2}-sin(t)+tg(t)]+ C=frac{1}{3}[e^{3x+2}-sin(3x)+tg(3x)]+ C =frac{1}{3}[e^{t+2}-sin(t)+tg(t)]+ C=frac{1}{3}[e^{3x+2}-sin(3x)+tg(3x)]+ C](https://tex.z-dn.net/?f=%3Dfrac%7B1%7D%7B3%7D%5Be%5E%7Bt%2B2%7D-sin%28t%29%2Btg%28t%29%5D%2B+C%3Dfrac%7B1%7D%7B3%7D%5Be%5E%7B3x%2B2%7D-sin%283x%29%2Btg%283x%29%5D%2B+C)
2)![F(x)= intlimits {[2e^{4x+ frac{1}{3} }-3sin( frac{x}{5} )-ctg( frac{6x}{5} )}] } , dx = F(x)= intlimits {[2e^{4x+ frac{1}{3} }-3sin( frac{x}{5} )-ctg( frac{6x}{5} )}] } , dx =](https://tex.z-dn.net/?f=F%28x%29%3D+intlimits+%7B%5B2e%5E%7B4x%2B+frac%7B1%7D%7B3%7D+%7D-3sin%28+frac%7Bx%7D%7B5%7D+%29-ctg%28+frac%7B6x%7D%7B5%7D+%29%7D%5D+%7D+%2C+dx+%3D+)
![= 2* frac{1}{4} intlimits {e^{4x+ frac{1}{3} }}d(4x+ frac{1}{3} )-3*5 intlimits {sin( frac{x}{5} )} , d( frac{x}{5}) - frac{5}{6} intlimits {ctg( frac{6x}{5} )} , d( frac{6x}{5} ) = = 2* frac{1}{4} intlimits {e^{4x+ frac{1}{3} }}d(4x+ frac{1}{3} )-3*5 intlimits {sin( frac{x}{5} )} , d( frac{x}{5}) - frac{5}{6} intlimits {ctg( frac{6x}{5} )} , d( frac{6x}{5} ) =](https://tex.z-dn.net/?f=%3D+2%2A+frac%7B1%7D%7B4%7D++intlimits+%7Be%5E%7B4x%2B+frac%7B1%7D%7B3%7D+%7D%7Dd%284x%2B+frac%7B1%7D%7B3%7D+%29-3%2A5+intlimits+%7Bsin%28+frac%7Bx%7D%7B5%7D+%29%7D+%2C+d%28+frac%7Bx%7D%7B5%7D%29+-++frac%7B5%7D%7B6%7D+intlimits+%7Bctg%28+frac%7B6x%7D%7B5%7D+%29%7D+%2C+d%28+frac%7B6x%7D%7B5%7D+%29+%3D)
![=frac{e^{4x+ frac{1}{3} }}{2}+15cos( frac{x}{5} ) - frac{5}{6} intlimits { frac{cos(frac{6x}{5} )}{sin(frac{6x}{5} )} } , d( frac{6x}{5} ) = =frac{e^{4x+ frac{1}{3} }}{2}+15cos( frac{x}{5} ) - frac{5}{6} intlimits { frac{cos(frac{6x}{5} )}{sin(frac{6x}{5} )} } , d( frac{6x}{5} ) =](https://tex.z-dn.net/?f=%3Dfrac%7Be%5E%7B4x%2B+frac%7B1%7D%7B3%7D+%7D%7D%7B2%7D%2B15cos%28+frac%7Bx%7D%7B5%7D+%29+-++frac%7B5%7D%7B6%7D+intlimits+%7B+frac%7Bcos%28frac%7B6x%7D%7B5%7D+%29%7D%7Bsin%28frac%7B6x%7D%7B5%7D+%29%7D+%7D+%2C+d%28+frac%7B6x%7D%7B5%7D+%29+%3D)
![=frac{e^{4x+ frac{1}{3} }}{2}+15cos( frac{x}{5} ) - frac{5}{6} intlimits { frac{1}{sin(frac{6x}{5} )} } , d( sin(frac{6x}{5} )) = =frac{e^{4x+ frac{1}{3} }}{2}+15cos( frac{x}{5} ) - frac{5}{6} intlimits { frac{1}{sin(frac{6x}{5} )} } , d( sin(frac{6x}{5} )) =](https://tex.z-dn.net/?f=%3Dfrac%7Be%5E%7B4x%2B+frac%7B1%7D%7B3%7D+%7D%7D%7B2%7D%2B15cos%28+frac%7Bx%7D%7B5%7D+%29+-++frac%7B5%7D%7B6%7D+intlimits+%7B+frac%7B1%7D%7Bsin%28frac%7B6x%7D%7B5%7D+%29%7D+%7D+%2C+d%28+sin%28frac%7B6x%7D%7B5%7D+%29%29+%3D)
![=frac{e^{4x+ frac{1}{3} }}{2}+15cos( frac{x}{5} ) - frac{5}{6} ln|sin(frac{6x}{5} )|+C =frac{e^{4x+ frac{1}{3} }}{2}+15cos( frac{x}{5} ) - frac{5}{6} ln|sin(frac{6x}{5} )|+C](https://tex.z-dn.net/?f=%3Dfrac%7Be%5E%7B4x%2B+frac%7B1%7D%7B3%7D+%7D%7D%7B2%7D%2B15cos%28+frac%7Bx%7D%7B5%7D+%29+-++frac%7B5%7D%7B6%7D+ln%7Csin%28frac%7B6x%7D%7B5%7D+%29%7C%2BC)
3)![F(x)= intlimits {[3e^{3x- frac{1}{2} }+3cos( frac{x}{7} )+ frac{1}{cos^2( frac{x}{3} )}] } , dx = F(x)= intlimits {[3e^{3x- frac{1}{2} }+3cos( frac{x}{7} )+ frac{1}{cos^2( frac{x}{3} )}] } , dx =](https://tex.z-dn.net/?f=F%28x%29%3D+intlimits+%7B%5B3e%5E%7B3x-+frac%7B1%7D%7B2%7D+%7D%2B3cos%28+frac%7Bx%7D%7B7%7D+%29%2B+frac%7B1%7D%7Bcos%5E2%28+frac%7Bx%7D%7B3%7D+%29%7D%5D+%7D+%2C+dx+%3D+)
![= 3*frac{1}{3} intlimits {e^{3x- frac{1}{2}} , d(3x- frac{1}{2})+3* 7intlimits {cos( frac{x}{7} )} , d(frac{x}{7})+ 3intlimits { frac{1}{cos^2( frac{x}{3} )} } , d( frac{x}{3} ) = = 3*frac{1}{3} intlimits {e^{3x- frac{1}{2}} , d(3x- frac{1}{2})+3* 7intlimits {cos( frac{x}{7} )} , d(frac{x}{7})+ 3intlimits { frac{1}{cos^2( frac{x}{3} )} } , d( frac{x}{3} ) =](https://tex.z-dn.net/?f=%3D+3%2Afrac%7B1%7D%7B3%7D++intlimits+%7Be%5E%7B3x-+frac%7B1%7D%7B2%7D%7D+%2C+d%283x-+frac%7B1%7D%7B2%7D%29%2B3%2A+7intlimits+%7Bcos%28+frac%7Bx%7D%7B7%7D+%29%7D+%2C+d%28frac%7Bx%7D%7B7%7D%29%2B+3intlimits+%7B+frac%7B1%7D%7Bcos%5E2%28+frac%7Bx%7D%7B3%7D+%29%7D+%7D+%2C+d%28+frac%7Bx%7D%7B3%7D+%29+%3D)
![= e^{3x- frac{1}{2}}+21sin( frac{x}{7} )}+ 3tg^2( frac{x}{3} ) = e^{3x- frac{1}{2}}+21sin( frac{x}{7} )}+ 3tg^2( frac{x}{3} )](https://tex.z-dn.net/?f=%3D+e%5E%7B3x-+frac%7B1%7D%7B2%7D%7D%2B21sin%28+frac%7Bx%7D%7B7%7D+%29%7D%2B+3tg%5E2%28+frac%7Bx%7D%7B3%7D+%29)
4)![F(x)= 2intlimits {x^{ frac{1}{3} }} , dx -3 intlimits {sin(6x-1)} , dx + intlimits {cos(2x+ frac{pi}{4}) } , dx F(x)= 2intlimits {x^{ frac{1}{3} }} , dx -3 intlimits {sin(6x-1)} , dx + intlimits {cos(2x+ frac{pi}{4}) } , dx](https://tex.z-dn.net/?f=F%28x%29%3D+2intlimits+%7Bx%5E%7B+frac%7B1%7D%7B3%7D+%7D%7D+%2C+dx+-3+intlimits+%7Bsin%286x-1%29%7D+%2C+dx+%2B+intlimits+%7Bcos%282x%2B+frac%7Bpi%7D%7B4%7D%29+%7D+%2C+dx+)
![=2 *frac{x^{ 1+frac{1}{3} }}{1+ frac{1}{3} } -3* frac{1}{6} intlimits {sin(6x-1)} , d(6x-1) + frac{1}{2} intlimits {cos(2x+ frac{pi}{4}) } , d(2x+ frac{pi}{4}) =2 *frac{x^{ 1+frac{1}{3} }}{1+ frac{1}{3} } -3* frac{1}{6} intlimits {sin(6x-1)} , d(6x-1) + frac{1}{2} intlimits {cos(2x+ frac{pi}{4}) } , d(2x+ frac{pi}{4})](https://tex.z-dn.net/?f=%3D2+%2Afrac%7Bx%5E%7B+1%2Bfrac%7B1%7D%7B3%7D+%7D%7D%7B1%2B+frac%7B1%7D%7B3%7D+%7D++-3%2A+frac%7B1%7D%7B6%7D++intlimits+%7Bsin%286x-1%29%7D+%2C+d%286x-1%29+%2B+frac%7B1%7D%7B2%7D++intlimits+%7Bcos%282x%2B+frac%7Bpi%7D%7B4%7D%29+%7D+%2C+d%282x%2B+frac%7Bpi%7D%7B4%7D%29+)
![= frac{3}{2}x^{ frac{4}{3}} + frac{1}{2} cos(6x-1)+ frac{1}{2} sin(2x+ frac{pi}{4}) } = frac{3}{2}x^{ frac{4}{3}} + frac{1}{2} cos(6x-1)+ frac{1}{2} sin(2x+ frac{pi}{4}) }](https://tex.z-dn.net/?f=%3D++frac%7B3%7D%7B2%7Dx%5E%7B+frac%7B4%7D%7B3%7D%7D++%2B+frac%7B1%7D%7B2%7D++cos%286x-1%29%2B+frac%7B1%7D%7B2%7D+sin%282x%2B+frac%7Bpi%7D%7B4%7D%29+%7D)
5)![F(x)= 5intlimits {x^{ frac{1}{4} }} , dx +2 intlimits {sin(5x+4)} , dx + intlimits {ctg(frac{3x+1}{4}) } , dx F(x)= 5intlimits {x^{ frac{1}{4} }} , dx +2 intlimits {sin(5x+4)} , dx + intlimits {ctg(frac{3x+1}{4}) } , dx](https://tex.z-dn.net/?f=F%28x%29%3D+5intlimits+%7Bx%5E%7B+frac%7B1%7D%7B4%7D+%7D%7D+%2C+dx+%2B2+intlimits+%7Bsin%285x%2B4%29%7D+%2C+dx+%2B+intlimits+%7Bctg%28frac%7B3x%2B1%7D%7B4%7D%29+%7D+%2C+dx+)
![F(x)= 4x^{ frac{5}{4} } - frac{2}{5} cos(5x+4)} + frac{4}{3} ln|sin(frac{3x+1}{4}))| F(x)= 4x^{ frac{5}{4} } - frac{2}{5} cos(5x+4)} + frac{4}{3} ln|sin(frac{3x+1}{4}))|](https://tex.z-dn.net/?f=F%28x%29%3D+4x%5E%7B+frac%7B5%7D%7B4%7D+%7D+-+frac%7B2%7D%7B5%7D+cos%285x%2B4%29%7D+%2B+frac%7B4%7D%7B3%7D+ln%7Csin%28frac%7B3x%2B1%7D%7B4%7D%29%29%7C)
6)![frac{3}{ sqrt{5} } intlimits {x^{ frac{1}{2} }} , dx -6 intlimits {cos( frac{2x-1}{3} )} , d(frac{2x-1}{3} )-4 intlimits {cos( frac{x}{4} - frac{pi}{2} )} , d(frac{x}{4} - frac{pi}{2}) = frac{3}{ sqrt{5} } intlimits {x^{ frac{1}{2} }} , dx -6 intlimits {cos( frac{2x-1}{3} )} , d(frac{2x-1}{3} )-4 intlimits {cos( frac{x}{4} - frac{pi}{2} )} , d(frac{x}{4} - frac{pi}{2}) =](https://tex.z-dn.net/?f=+frac%7B3%7D%7B+sqrt%7B5%7D+%7D+intlimits+%7Bx%5E%7B+frac%7B1%7D%7B2%7D+%7D%7D+%2C+dx++-6+intlimits+%7Bcos%28+frac%7B2x-1%7D%7B3%7D+%29%7D+%2C+d%28frac%7B2x-1%7D%7B3%7D+%29-4+intlimits+%7Bcos%28+frac%7Bx%7D%7B4%7D+-+frac%7Bpi%7D%7B2%7D+%29%7D+%2C+d%28frac%7Bx%7D%7B4%7D+-+frac%7Bpi%7D%7B2%7D%29++%3D)
![frac{2}{ sqrt{5} } x^{ frac{3}{2} } -6 sin( frac{2x-1}{3} )-4sin( frac{x}{4} - frac{pi}{2} ) frac{2}{ sqrt{5} } x^{ frac{3}{2} } -6 sin( frac{2x-1}{3} )-4sin( frac{x}{4} - frac{pi}{2} )](https://tex.z-dn.net/?f=+frac%7B2%7D%7B+sqrt%7B5%7D+%7D+x%5E%7B+frac%7B3%7D%7B2%7D+%7D++-6+sin%28+frac%7B2x-1%7D%7B3%7D+%29-4sin%28+frac%7Bx%7D%7B4%7D+-+frac%7Bpi%7D%7B2%7D+%29)
7)![frac{2}{ sqrt[3]{3} } intlimits {x^{ frac{1}{3} }} , dx +4 intlimits {sin( frac{5x-3}{4} )} , d(frac{5x-3}{4} )+ frac{1}{3} intlimits { frac{1}{3x-1} } , d(3x-1) = frac{2}{ sqrt[3]{3} } intlimits {x^{ frac{1}{3} }} , dx +4 intlimits {sin( frac{5x-3}{4} )} , d(frac{5x-3}{4} )+ frac{1}{3} intlimits { frac{1}{3x-1} } , d(3x-1) =](https://tex.z-dn.net/?f=frac%7B2%7D%7B++sqrt%5B3%5D%7B3%7D++%7D+intlimits+%7Bx%5E%7B+frac%7B1%7D%7B3%7D+%7D%7D+%2C+dx+%2B4+intlimits+%7Bsin%28+frac%7B5x-3%7D%7B4%7D+%29%7D+%2C+d%28frac%7B5x-3%7D%7B4%7D+%29%2B+frac%7B1%7D%7B3%7D+intlimits+%7B+frac%7B1%7D%7B3x-1%7D+%7D+%2C+d%283x-1%29+++%3D)
![frac{ sqrt[3]{9} }{ 2}* x^{ frac{4}{3} }} -4* cos( frac{5x-3}{4} )+ frac{1}{3}* ln|3x-1| frac{ sqrt[3]{9} }{ 2}* x^{ frac{4}{3} }} -4* cos( frac{5x-3}{4} )+ frac{1}{3}* ln|3x-1|](https://tex.z-dn.net/?f=frac%7B+sqrt%5B3%5D%7B9%7D++%7D%7B+2%7D%2A+x%5E%7B+frac%7B4%7D%7B3%7D+%7D%7D+-4%2A+cos%28+frac%7B5x-3%7D%7B4%7D+%29%2B+frac%7B1%7D%7B3%7D%2A+ln%7C3x-1%7C)
8)![F(x)= intlimits {[frac{3}{ sqrt[3]{2x-1} } + frac{2}{1-5x}+e^{frac{2x-1}{5}} ]} , dx = F(x)= intlimits {[frac{3}{ sqrt[3]{2x-1} } + frac{2}{1-5x}+e^{frac{2x-1}{5}} ]} , dx =](https://tex.z-dn.net/?f=F%28x%29%3D++intlimits+%7B%5Bfrac%7B3%7D%7B+sqrt%5B3%5D%7B2x-1%7D+%7D+%2B+frac%7B2%7D%7B1-5x%7D%2Be%5E%7Bfrac%7B2x-1%7D%7B5%7D%7D+%5D%7D+%2C+dx+%3D)
![=3 intlimits {frac{1}{ sqrt[3]{2x-1} } , dx + 2 intlimits {frac{1}{1-5x}} , dx + intlimits {e^{frac{2x-1}{5}} } , dx= =3 intlimits {frac{1}{ sqrt[3]{2x-1} } , dx + 2 intlimits {frac{1}{1-5x}} , dx + intlimits {e^{frac{2x-1}{5}} } , dx=](https://tex.z-dn.net/?f=%3D3+intlimits+%7Bfrac%7B1%7D%7B+sqrt%5B3%5D%7B2x-1%7D+%7D+%2C+dx++%2B+2+intlimits+%7Bfrac%7B1%7D%7B1-5x%7D%7D+%2C+dx+%2B+intlimits+%7Be%5E%7Bfrac%7B2x-1%7D%7B5%7D%7D+%7D+%2C+dx%3D)
![= frac{3}{2} intlimits {(2x-1)^ {-frac{1}{3} } , d(2x-1) - frac{2}{5} intlimits {frac{1}{5x-1}} , d(5x-1) + frac{5}{2} intlimits {e^{frac{2x-1}{5}} } , d(frac{2x-1}{5}} ) = frac{3}{2} intlimits {(2x-1)^ {-frac{1}{3} } , d(2x-1) - frac{2}{5} intlimits {frac{1}{5x-1}} , d(5x-1) + frac{5}{2} intlimits {e^{frac{2x-1}{5}} } , d(frac{2x-1}{5}} )](https://tex.z-dn.net/?f=%3D+frac%7B3%7D%7B2%7D++intlimits+%7B%282x-1%29%5E+%7B-frac%7B1%7D%7B3%7D+%7D+%2C+d%282x-1%29++-++frac%7B2%7D%7B5%7D++intlimits+%7Bfrac%7B1%7D%7B5x-1%7D%7D+%2C+d%285x-1%29+%2B++frac%7B5%7D%7B2%7D+intlimits+%7Be%5E%7Bfrac%7B2x-1%7D%7B5%7D%7D+%7D+%2C+d%28frac%7B2x-1%7D%7B5%7D%7D+%29)
![= (2x-1)^ {frac{2}{3} } - frac{2}{5} ln|5x-1| + frac{5}{2} e^{frac{2x-1}{5}} = (2x-1)^ {frac{2}{3} } - frac{2}{5} ln|5x-1| + frac{5}{2} e^{frac{2x-1}{5}}](https://tex.z-dn.net/?f=%3D+%282x-1%29%5E+%7Bfrac%7B2%7D%7B3%7D+%7D++-++frac%7B2%7D%7B5%7D++ln%7C5x-1%7C+%2B++frac%7B5%7D%7B2%7D+e%5E%7Bfrac%7B2x-1%7D%7B5%7D%7D)
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