1.
₀∫²π/₃(2*sin(π/3+x)dx=-2*cos(π/3+x) ₀|²π/₃=-2*cos(π/3+2π/3)-(-2*cos(π/3+0)=
=-2*cos(π)+2*cos(π/3)=-2*(-1)+2*(1/2)=2+1=3.
2.
√(3x+1)=2x ОДЗ: 3x+1≥0 x≥-1/3 2x≥0 x≥0 ⇒ x∈[0;+∞)
(√3x+1))²=(2x)²
3x+1=4x²
4x²-3x-1=0 D=25
x₁=1 x₂=-1/4 ∉ОДЗ.
Ответ: x=1.
3.
f(x)=x⁴-2x²+3
f`(x)=4x³-4x=0
4x*(x²-1)=0
4x=0
x₁=0
x²-1=0
x²=1
x₂=1 x₃=-1
f(-4)=(-4)⁴-2*(-4)²+3=256-32+3=227=ymax
f(-1)=(-1)⁴-2*(-1)²+3=1-2+3=2=ymin
f(0)=0⁴-2*0²+3=3
f(1)=1⁴-2*1+3=1-2+3=2=ymin
f(3)=3⁴-2*3²+3=81-18+3=66.
4.
log₅(25/∛5)+log₇∛49=log₅(5²/∛5)+log₇∛7²=log₅(5^(2-1/3)+log₇7²/³=
log₅5⁵/³+log7²/³=5/3+2/3=7/3=2¹/₃.
5.
log₄x+log₄y=1
y-2x=7 ОДЗ: x.>0 y>0
y=2x+7
log₄x+log₄(2x+7)=1
log₄(x*(2x+7))=log₄4
2x²+7x=4
2x²+7x-4=0 D=81
x₁=0,5 x₂=-4 ∉ ОДЗ
y=2*0,5+7=1+7=8
Ответ: x=0,5 y=8.