.........................................
4-х/7=х/9 /*63
252-9х=7х
-9х-7х= -252
-16х= -252
х= -252:(-16)
х=15,75
Так как (cosx)`=-sinx,
то это интеграл от степенной функции
Решается методом замены переменной
u= cosx
du=-sin dx, значит sinx dx=-du
![= \int {u ^{3} } \,(-du)=- \frac{u^{4}}{4}+C=- \frac{cos^{4}x }{4}+C](https://tex.z-dn.net/?f=%3D%20%5Cint%20%7Bu%20%5E%7B3%7D%20%7D%20%5C%2C%28-du%29%3D-%20%5Cfrac%7Bu%5E%7B4%7D%7D%7B4%7D%2BC%3D-%20%5Cfrac%7Bcos%5E%7B4%7Dx%20%7D%7B4%7D%2BC%20%20%20)
Проверка
![(- \frac{cos^{4}x }{4}+C)`=(- \frac{cos^{4}x }{4})`+(C)` =- \frac{1}{4}(cos ^{4}x)`+0= \\ \\ =- \frac{1}{4}\cdot 4 cos^{3}x\cdot (cosx)`=-cos ^{3}x\cdot (-sinx)=sinx\cdot cos ^{3}x](https://tex.z-dn.net/?f=%28-%20%5Cfrac%7Bcos%5E%7B4%7Dx%20%7D%7B4%7D%2BC%29%60%3D%28-%20%5Cfrac%7Bcos%5E%7B4%7Dx%20%7D%7B4%7D%29%60%2B%28C%29%60%20%3D-%20%5Cfrac%7B1%7D%7B4%7D%28cos%20%5E%7B4%7Dx%29%60%2B0%3D%20%5C%5C%20%20%5C%5C%20%3D-%20%5Cfrac%7B1%7D%7B4%7D%5Ccdot%204%20cos%5E%7B3%7Dx%5Ccdot%20%28cosx%29%60%3D-cos%20%5E%7B3%7Dx%5Ccdot%20%28-sinx%29%3Dsinx%5Ccdot%20cos%20%5E%7B3%7Dx%20%20%20%20%20%20%20%20)
А я, пожалуй, остальные добью.
г)
![\int \cos^{3/4}(x+3)\sin(x+3)\ dx=-\int \cos^{3/4}(x+3)\ d\cos(x+3)=\\=-\int u^{3/4}du=-\frac47 u^{7/4}+C=-\frac47 \cos^{7/4}(x+3)+C](https://tex.z-dn.net/?f=%5Cint%20%5Ccos%5E%7B3%2F4%7D%28x%2B3%29%5Csin%28x%2B3%29%5C%20dx%3D-%5Cint%20%5Ccos%5E%7B3%2F4%7D%28x%2B3%29%5C%20d%5Ccos%28x%2B3%29%3D%5C%5C%3D-%5Cint%20u%5E%7B3%2F4%7Ddu%3D-%5Cfrac47%20u%5E%7B7%2F4%7D%2BC%3D-%5Cfrac47%20%5Ccos%5E%7B7%2F4%7D%28x%2B3%29%2BC)
д)
![\int5x^4\sin5x^5dx=\frac15\int\sin5x^5d(5x^5)=\frac15\int\sin u\,du=-\frac15\cos u+C\\=-\frac15\cos5x^5+C](https://tex.z-dn.net/?f=%5Cint5x%5E4%5Csin5x%5E5dx%3D%5Cfrac15%5Cint%5Csin5x%5E5d%285x%5E5%29%3D%5Cfrac15%5Cint%5Csin%20u%5C%2Cdu%3D-%5Cfrac15%5Ccos%20u%2BC%5C%5C%3D-%5Cfrac15%5Ccos5x%5E5%2BC)
Почленное интегрирование
а)
![\int(5\cos4x+\frac2{(x-3)^2+1}-\frac1{x-1})dx=\int5\cos4x\,dx+\int \frac2{(x-3)^2+1}dx-\\-\int\frac{dx}{x-1}=\frac54\sin4x+2\,\mathrm{arctg}\,(x-3)-\ln|x-1|+C](https://tex.z-dn.net/?f=%5Cint%285%5Ccos4x%2B%5Cfrac2%7B%28x-3%29%5E2%2B1%7D-%5Cfrac1%7Bx-1%7D%29dx%3D%5Cint5%5Ccos4x%5C%2Cdx%2B%5Cint%20%5Cfrac2%7B%28x-3%29%5E2%2B1%7Ddx-%5C%5C-%5Cint%5Cfrac%7Bdx%7D%7Bx-1%7D%3D%5Cfrac54%5Csin4x%2B2%5C%2C%5Cmathrm%7Barctg%7D%5C%2C%28x-3%29-%5Cln%7Cx-1%7C%2BC)
б)
![\int(3\cos(x-3)-4e^{2x}+3x^4)dx=3\int\cos(x-3)\,dx-2\int 2e^{2x}dx+\\+3\int x^4\,dx=3\sin(x-3)-e^{2x}+\frac35x^5+C](https://tex.z-dn.net/?f=%5Cint%283%5Ccos%28x-3%29-4e%5E%7B2x%7D%2B3x%5E4%29dx%3D3%5Cint%5Ccos%28x-3%29%5C%2Cdx-2%5Cint%202e%5E%7B2x%7Ddx%2B%5C%5C%2B3%5Cint%20x%5E4%5C%2Cdx%3D3%5Csin%28x-3%29-e%5E%7B2x%7D%2B%5Cfrac35x%5E5%2BC)
в)
![\int(3^{4x}+2(3x)^{1/3}-\frac4{\sin^23x})dx=\frac{3^{4x}}{4\ln3}+\frac92(3x)^{4/3}+\frac43\,\mathrm{ctg}\,(3x)+C](https://tex.z-dn.net/?f=%5Cint%283%5E%7B4x%7D%2B2%283x%29%5E%7B1%2F3%7D-%5Cfrac4%7B%5Csin%5E23x%7D%29dx%3D%5Cfrac%7B3%5E%7B4x%7D%7D%7B4%5Cln3%7D%2B%5Cfrac92%283x%29%5E%7B4%2F3%7D%2B%5Cfrac43%5C%2C%5Cmathrm%7Bctg%7D%5C%2C%283x%29%2BC)