помогите пожалуйста решить)заранее спасибо=)
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Ответил mappku
0
19)
![int{sin^5xcos x}dx=int{sin^5x}d(sin x)=left|sin x=t;d(sin x)=dtright|=\
=int{t^5}dt=frac{1}{5+1}cdot t^{5+1}+C=frac16sin^6x+C;](https://tex.z-dn.net/?f=int%7Bsin%5E5xcos+x%7Ddx%3Dint%7Bsin%5E5x%7Dd%28sin+x%29%3Dleft%7Csin+x%3Dt%3Bd%28sin+x%29%3Ddtright%7C%3D%5C%0A%3Dint%7Bt%5E5%7Ddt%3Dfrac%7B1%7D%7B5%2B1%7Dcdot+t%5E%7B5%2B1%7D%2BC%3Dfrac16sin%5E6x%2BC%3B "int{sin^5xcos x}dx=int{sin^5x}d(sin x)=left|sin x=t;d(sin x)=dtright|=\
=int{t^5}dt=frac{1}{5+1}cdot t^{5+1}+C=frac16sin^6x+C;")
20)
![int{frac{dx}{arcsin^3xsqrt{1-x^2}}}=left|d(arcsin x=frac{dx}{sqrt{1-x^2}})right|=\
=int{frac{d(arcsin x)}{arcsin^3x}}=left|arcsin x=t; d(arcsin x)=dt right|=\
=int{frac{dt}{t^3}}=int{t^{-3}}dt=frac{1}{-3+1}cdot t^{-3+1}+C=-frac12t^{-2}+C=\
=-frac{1}{2t^2}+C=-frac{1}{2arcsin^2x}+C](https://tex.z-dn.net/?f=int%7Bfrac%7Bdx%7D%7Barcsin%5E3xsqrt%7B1-x%5E2%7D%7D%7D%3Dleft%7Cd%28arcsin+x%3Dfrac%7Bdx%7D%7Bsqrt%7B1-x%5E2%7D%7D%29right%7C%3D%5C%0A%3Dint%7Bfrac%7Bd%28arcsin+x%29%7D%7Barcsin%5E3x%7D%7D%3Dleft%7Carcsin+x%3Dt%3B+d%28arcsin+x%29%3Ddt+right%7C%3D%5C%0A%3Dint%7Bfrac%7Bdt%7D%7Bt%5E3%7D%7D%3Dint%7Bt%5E%7B-3%7D%7Ddt%3Dfrac%7B1%7D%7B-3%2B1%7Dcdot+t%5E%7B-3%2B1%7D%2BC%3D-frac12t%5E%7B-2%7D%2BC%3D%5C%0A%3D-frac%7B1%7D%7B2t%5E2%7D%2BC%3D-frac%7B1%7D%7B2arcsin%5E2x%7D%2BC "int{frac{dx}{arcsin^3xsqrt{1-x^2}}}=left|d(arcsin x=frac{dx}{sqrt{1-x^2}})right|=\
=int{frac{d(arcsin x)}{arcsin^3x}}=left|arcsin x=t; d(arcsin x)=dt right|=\
=int{frac{dt}{t^3}}=int{t^{-3}}dt=frac{1}{-3+1}cdot t^{-3+1}+C=-frac12t^{-2}+C=\
=-frac{1}{2t^2}+C=-frac{1}{2arcsin^2x}+C")
21)
![int{frac{ctg^7x}{sin^2x}}dx=left|d(ctg x)=dleft(frac{cos x}{sin x}right)=frac{cos'xsin x-cos xsin'x}{sin^2x}dx=right|\
=left|frac{-cos^2x-sin^2x}{sin^2x}dx=-frac{dx}{sin^2x} frac{dx}{sin^2x}=-d(ctgx)right|=\
=-int{ctg^7x }d(ctgx)=left|ctgx=t; d(ctgx)=dtright|=\
=-int{t^7}dt=-frac{1}{7+1}cdot t^{7+1}+C=-frac{t^8}{8}+C=-frac18ctg^8x+C](https://tex.z-dn.net/?f=int%7Bfrac%7Bctg%5E7x%7D%7Bsin%5E2x%7D%7Ddx%3Dleft%7Cd%28ctg+x%29%3Ddleft%28frac%7Bcos+x%7D%7Bsin+x%7Dright%29%3Dfrac%7Bcos%27xsin+x-cos+xsin%27x%7D%7Bsin%5E2x%7Ddx%3Dright%7C%5C%0A%3Dleft%7Cfrac%7B-cos%5E2x-sin%5E2x%7D%7Bsin%5E2x%7Ddx%3D-frac%7Bdx%7D%7Bsin%5E2x%7D++frac%7Bdx%7D%7Bsin%5E2x%7D%3D-d%28ctgx%29right%7C%3D%5C%0A%3D-int%7Bctg%5E7x+%7Dd%28ctgx%29%3Dleft%7Cctgx%3Dt%3B+d%28ctgx%29%3Ddtright%7C%3D%5C%0A%3D-int%7Bt%5E7%7Ddt%3D-frac%7B1%7D%7B7%2B1%7Dcdot+t%5E%7B7%2B1%7D%2BC%3D-frac%7Bt%5E8%7D%7B8%7D%2BC%3D-frac18ctg%5E8x%2BC "int{frac{ctg^7x}{sin^2x}}dx=left|d(ctg x)=dleft(frac{cos x}{sin x}right)=frac{cos'xsin x-cos xsin'x}{sin^2x}dx=right|\
=left|frac{-cos^2x-sin^2x}{sin^2x}dx=-frac{dx}{sin^2x} frac{dx}{sin^2x}=-d(ctgx)right|=\
=-int{ctg^7x }d(ctgx)=left|ctgx=t; d(ctgx)=dtright|=\
=-int{t^7}dt=-frac{1}{7+1}cdot t^{7+1}+C=-frac{t^8}{8}+C=-frac18ctg^8x+C")
22)
![int{(3x+4)}e^{3x}dx=left|int{udv}-ucdot v-int{vdu}right|=\
left|u=(3x+4); du=3dxright|\
left|dv=e^{3x}dx; v=frac13e^{3x}right|\
=frac13(3x+4)e^{3x}-int{frac13e^{3x}3dx}=(x+frac43)e^3x-int{e^{3x}}dx=\
=(x+frac43)e^{3x}-frac13e^{3x}+C=(x+frac43-frac13)e^{3x}+C=\
=(x+1)e^{3x}+C.](https://tex.z-dn.net/?f=int%7B%283x%2B4%29%7De%5E%7B3x%7Ddx%3Dleft%7Cint%7Budv%7D-ucdot+v-int%7Bvdu%7Dright%7C%3D%5C%0Aleft%7Cu%3D%283x%2B4%29%3B+++du%3D3dxright%7C%5C%0Aleft%7Cdv%3De%5E%7B3x%7Ddx%3B++v%3Dfrac13e%5E%7B3x%7Dright%7C%5C%0A%3Dfrac13%283x%2B4%29e%5E%7B3x%7D-int%7Bfrac13e%5E%7B3x%7D3dx%7D%3D%28x%2Bfrac43%29e%5E3x-int%7Be%5E%7B3x%7D%7Ddx%3D%5C%0A%3D%28x%2Bfrac43%29e%5E%7B3x%7D-frac13e%5E%7B3x%7D%2BC%3D%28x%2Bfrac43-frac13%29e%5E%7B3x%7D%2BC%3D%5C%0A%3D%28x%2B1%29e%5E%7B3x%7D%2BC. "int{(3x+4)}e^{3x}dx=left|int{udv}-ucdot v-int{vdu}right|=\
left|u=(3x+4); du=3dxright|\
left|dv=e^{3x}dx; v=frac13e^{3x}right|\
=frac13(3x+4)e^{3x}-int{frac13e^{3x}3dx}=(x+frac43)e^3x-int{e^{3x}}dx=\
=(x+frac43)e^{3x}-frac13e^{3x}+C=(x+frac43-frac13)e^{3x}+C=\
=(x+1)e^{3x}+C.")
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