![\int\limits^{ \frac{2 \pi }{3} }_0 {2sin( \frac{ \pi }{3}+x) } \, dx=-2cos( \frac{ \pi }{3}+x)| ^{ \frac{2 \pi }{3} }_0 =-2\cdot(cos( \frac{ \pi }{3}+ \frac{2 \pi }{3})-cos \frac{ \pi }{3})= \\ \\ =-2\cdotcos \pi +2cos \frac{ \pi }{3}=-2\cdot(-1)+2\cdot \frac{1}{2}=3](https://tex.z-dn.net/?f=+%5Cint%5Climits%5E%7B+%5Cfrac%7B2+%5Cpi+%7D%7B3%7D+%7D_0+%7B2sin%28+%5Cfrac%7B+%5Cpi+%7D%7B3%7D%2Bx%29+%7D+%5C%2C+dx%3D-2cos%28+%5Cfrac%7B+%5Cpi+%7D%7B3%7D%2Bx%29%7C+%5E%7B+%5Cfrac%7B2+%5Cpi+%7D%7B3%7D+%7D_0+%3D-2%5Ccdot%28cos%28+%5Cfrac%7B+%5Cpi+%7D%7B3%7D%2B+%5Cfrac%7B2+%5Cpi+%7D%7B3%7D%29-cos+%5Cfrac%7B+%5Cpi+%7D%7B3%7D%29%3D+%5C%5C++%5C%5C+%3D-2%5Ccdotcos+%5Cpi+%2B2cos+%5Cfrac%7B+%5Cpi+%7D%7B3%7D%3D-2%5Ccdot%28-1%29%2B2%5Ccdot+%5Cfrac%7B1%7D%7B2%7D%3D3++)
f`(x)=4x³-4x
f`(x)=0
4x³-4x=0
4x(x²-1)=0
4x(x-1)(x+1)=0
x=-1;x=0;x=1
[-4]_____-____ (-1) _+_ (0) _-_ (1) ___+___ [3]
x=-1 - точка минимума
х=0 - точка максимума
х=1- точка минимума
f(-4)=(-4)⁴-2·(-4)²+3=256-32+3=227
f(-1)=(-1)⁴-2·(-1)²+3=2
f(0)=0⁴-2·0²+3=3
f(1)=1⁴-2·1²+3=2
f(3)=3⁴-2·3²+3=66
О т в е т. 227- наибольшее, 2 - наименьшее.
![log_{5} \frac{25}{ \sqrt[3]{5} }= log_{5}5^{2- \frac{1}{3}}=log_{5}5^{ \frac{5}{3}} = \frac{5}{3} \\ \\ log_{7} \sqrt[3]{49}=log_{7}7^{ \frac{2}{3} }= \frac{2}{3}](https://tex.z-dn.net/?f=log_%7B5%7D+%5Cfrac%7B25%7D%7B+%5Csqrt%5B3%5D%7B5%7D+%7D%3D+log_%7B5%7D5%5E%7B2-+%5Cfrac%7B1%7D%7B3%7D%7D%3Dlog_%7B5%7D5%5E%7B+%5Cfrac%7B5%7D%7B3%7D%7D+%3D+%5Cfrac%7B5%7D%7B3%7D+%5C%5C++++%5C%5C+log_%7B7%7D+%5Csqrt%5B3%5D%7B49%7D%3Dlog_%7B7%7D7%5E%7B+%5Cfrac%7B2%7D%7B3%7D+%7D%3D+%5Cfrac%7B2%7D%7B3%7D++)
О т в е т. (5/3)+(2/3)=7/3
![\int\limits^{ 2 \pi }_0 {sin(x+ \frac{ \pi }{3}) } \, dx=-cos(x+ \frac{ \pi }{3})| ^{ 2 \pi }_0 =-(cos( 2 \pi +\frac{ \pi }{3})-cos(0+ \frac{ \pi }{3}))=0](https://tex.z-dn.net/?f=+%5Cint%5Climits%5E%7B+2+%5Cpi+%7D_0+%7Bsin%28x%2B+%5Cfrac%7B+%5Cpi+%7D%7B3%7D%29+%7D+%5C%2C+dx%3D-cos%28x%2B+%5Cfrac%7B+%5Cpi+%7D%7B3%7D%29%7C+%5E%7B+2+%5Cpi+%7D_0+%3D-%28cos%28+2+%5Cpi+%2B%5Cfrac%7B+%5Cpi+%7D%7B3%7D%29-cos%280%2B+%5Cfrac%7B+%5Cpi+%7D%7B3%7D%29%29%3D0)
Используем формулу разложения суммы кубов:
![x^3+y^3+2xy (x+y) = (x+y) (x^2 - xy + y^2) + 2xy (x+y) = \\ \\ = (x+y)(x^2 - xy + y^2+ 2xy) = (x+y)(x^2 + xy + y^2)](https://tex.z-dn.net/?f=x%5E3%2By%5E3%2B2xy+%28x%2By%29+%3D+%28x%2By%29+%28x%5E2+-+xy+%2B+y%5E2%29+%2B+2xy+%28x%2By%29+%3D++%5C%5C++%5C%5C+%3D+%28x%2By%29%28x%5E2+-+xy+%2B+y%5E2%2B+2xy%29+%3D+%28x%2By%29%28x%5E2+%2B+xy+%2B+y%5E2%29+)
Лови 4x + 3a) - (5x - 3a) = 4x + 3a - 5x +3a = 6a - x