lg(x^2+2)=lgx+lg3的解是

设u（x）=(x^2+1)(x^2+2)…(x^2+2009)
所以f(x)=xu(x)
f'(x)=x'u(x)+xu'(x)=u(x)+xu'(x)
x=0时,xu'(x)=0*u‘（0）=0
所以f'(0)=u(0)=1*2*3*……*2009=2009!

∫(x^2+2^x+e^x)dx
=x^3/3+2^x/ln(2)+e^x+c

x²+2(a+b+c)x+3(ab+bc+ca)=0
判别式△=[2(a+b+c)]²-4×1×[3(ab+bc+ca)]
=4(a²+b²+c²+2ab+2bc+2ca)-12(ab+bc+ca)
=4a²+4b²+4c²-4ab-4bc-4ca
=2(a²-2ab+b²)+2(b²-2bc+c²)+2(c²-2ca+a²)
=2(a-b)²+2(b-c)²+2(c-a)²
平方项恒非负,(a-b)²≥0,2(a-b)²≥0,同理,2(b-c)²≥0 2(c-a)²≥0
2(a-b)²+2(b-c)²+2(c-a)²≥0 △≥0,方程恒有实根.
当a=b=c时,方程有两相等实根,当a、b、c不全相等时,方程有两不等实根.
x²+2(a+b+1)x+3(a+1)(b+1)-3=0
判别式△=[2(a+b+1)]²-4×1×[3(a+1)(b+1)-3]
=4(a²+b²+1+2ab+2a+2b)-12(ab+a+b+1)+12
=4a²+4b²+8ab+8a+8b+4-12ab-12a-12b-12+12
=4a²+4b²-4ab-4a-4b+4
=(2a²-4a+2)+(2b²-4b+2)+(2a²-4ab+2b²)
=2(a-1)²+2(b-1)²+2(a-b)²
平方项恒非负,(a-1)²≥0,2(a-1)²≥0,同理,2(b-1)²≥0 2(a-b)²≥0
2(a-1)²+2(b-1)²+2(a-b)²≥0 △≥0,方程恒有实根.
当a=b=1时,方程有两相等实根,其余情况下,方程有两不等实根.

(x^4+3x^3+5x^2+7x+9)/(x^2+2)
=X^6+3X^5+5X^4+7X^3+9X^2+2x^4+6x^3+10x^2+14x+18
=X^6+3X^5+7X^4+13X^3+19X^2+14x+18