Task/26025958
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Условия не перепишу
1.
а)
F(x) =8x - x⁵
F'(x) =(8x-x⁵)' =(8x) ' - (x⁵) '= 8 -5x⁴ = f(x) .
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б)
F'(x) =(2tgx-3x)' =(2tgx) '=2(tgx)' - 3(x) '= 2/cos²x -3 = f(x) .
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2.
а)
F(x)=∫(3cosx-3/x⁴)dx =3∫cosxdx -3∫x⁻⁴ dx=3sinx -3x⁻³/(-3)+C=3sinx+1/x³+C.
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б)F(x)=∫2cos2xsin2xdx= ∫sin4xdx = - cos4x + C.
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3.
F(x)=∫ ( (1/3)*cos3x -1)dx =(1/9)*sin3x -x +C
т.к. график F(x) проходит через точку M(π/3; π/6) ,то :
π/6 =(1/9)*sin(3*π/3) - π/3 +C ⇔π/6 =(1/9)*sinπ -π/3 +C⇔π/6 =0-π/3+C⇒
C=π/2 и F(x)=(1/9)*sin3x - x + π/2 .