3x^4 +4x^3+7x^2+4x +3

∫x^11(2+3x^4)^(1/2)dx=(1/12)∫(2+3x^4)^(1/2)dx^12
(3x^4/2)=t
原式=(9/32)∫(2+2t)^(1/2)dt^3=(9/32)t^3(2+2t)^(1/2)-(27√2/(64)∫(t^3+t^2-t^2-t+t)dt/(1+t)^(1/2)
=(9/32)t^3(2+2t)^(1/2)-(27√2/(64)∫t^2(1+t)^(1/2)dt+(27√2/(64)∫t(1+t)^(1/2)dt-(27√2/(64)∫(t+1)^(1/2)dt+(27√2/(64)∫dt/(1+t)^(1/2)
(27√2/(64)∫t^2(1+t)^(1/2)dt=(9/64)∫(2+2t)^(1/2)dt^3
(27/64)∫(2+2t)^(1/2)dt^3=(9/32)t^3(2+2t)^(1/2)+(27√2/128)∫(1+t)^(1/2)dt^2-(9√2/32)(1+t)^(3/2)+(27√2/32)(1+t)^(1/2)=(9/32)t^3(2+2t)^(1/2)+(27√2/128)t^2(1+t)^(1/2)-(27√2/256)∫(t^2+t-t-1+1)dt/(1+t)^(1/2)-(9√2/32)(1+t)^(3/2)+(27√2/32)(1+t)^(1/2)
=(9/32)t^3(2+2t)^(1/2)+(27√2/128)t^2(1+t)^(1/2)-(27√2/640)(1+t)^(5/2)+(9√2/128)(1+t)^(3/2)+(9√2/128)(1+t)^(3/2)-(27√2/128)(1+t)^(1/2)-(9√2/32)(1+t)^(3/2)+(27√2/32)(1+t)^(1/2)
=9/32t^3(2+2t)^(1/2)+(27√2/128)t^2(1+t)^(1/2)-(27√2/640)(1+t)^(5/2)-(9√2/64)(1+t)^(3/2)
原式=(9/32)∫(2+2t)^2dt^3=(27/64)t^3(2+2t)^(1/2)+(81√2/256)t^2(1+t)^(1/2)-(81√2/1280)(1+t)^(5/2)-(27√2/128)(1+t)^(3/2)
=(27/64)(27/8)x^12(2+3x^4)^(1/2)+(81/256)(9/4)x^8(2+3x^4)^(1/2)-(81√2/(1280)∫(1+3x^4/2)^(5/2)-(27√2/128)(1+3x^4/2)^(3/2)

当x不大于2时候 f(x)=x^4(x-2)-x^3(x-2)+4x^2(x-2)+8x(x-2)+6(x-2)+17
f(x)/g(x)=x^4-x^3+4x^2+8x+6+17/(x-2)
方法二：可令x-2=t,则x=2+t 代入f（x）转换为t的函数 再计算

令t=3x^4
则dt=12x³dx
所以∫x³/(1-3x^4)dx
=(1/12)∫12x³/(1-3x^4) dx
=(1/12)∫1/(1-t) dt
=(1/12)ln|1-t|+C

∫(x^11)/(x^8+3x^4+2)dx
=1/4*∫(x^8)/(x^8+3x^4+2)dx^4
t=x^4
原积分=1/4*∫t^2/(t^2+3t+2)dt=1/4*∫(t^2+3t+2-3t-3+1)/(t+1)(t+2) dt
=1/4*∫dt-1/4*∫3/(t+2)dt+1/4*∫[1/(t+1)-1/(t+2)]dt
=1/4t-3/4*ln(t+2)+1/4*ln(t+1)/(t+2)+c
=1/4*x^4-1/4*ln(x^4+1)-ln(x^4+2)+c

∫ （3x^4+3x²+1）/（x²+1） dx
=∫ 3x²+ 1/（x²+1） dx
=x^3+arctanx+C

x^2+x+3=0
x^2+x=-3
x^5+3x^4+2x^3+2x^2-10x+5
=x^5+x^4+2x^4+2x^3+2x^2-10x+5
=x^3(x^2+x)+2x^4+2x^3+2x^2-10x+5
=-3x^3+2x^4+2x^3+2x^2-10x+5
=2x^4-x^3+2x^2-10x+5
=2x^4+2x^3-3x^3+2x^2-10x+5
=2x^2(x^2+x)-3x^3+2x^2-10x+5
=-6x^2-3x^3+2x^2-10x+5
=-3x^3-4x^2-10x+5
=-3x^3-3x^2-x^2-10x+5
=-3x(x^2+x)-x^2-10x+5
=9x-x^2-10x+5
=-x^2-x+5
=-(x^2+x)+5
=3+5
=8

（1） -(-2x^3)^2·2x^2+(-3x^4)^2
=-8x^8+9x^8
=x^8

（2） -1/2(-2xyz^2)^3
=-1/2*(-8x^3y^3z^6)
=4x^3y^3z^6

原式=∫(3x^4+3x^2-2x^2-2+2)/(x^2+1)dx
=∫[3x^2-2+2/(x^2+1)]dx
=x^3-2x+2arctanx+C

有两个实根:x = 2,x = -1/3

3x^4+4x^2-7=0
(3x^2+7)(x^2-1)=0
(3x^2+7)(x+1)(x-1)=0
x+1=0或x-1=0
x1=-1 x2=1

这两个方程都有个特点：高低项系数是对称相等或互为相反数的.
可用以下代换方法解得.
令t=x-1/x,则有t^2+2=x^2+1/x^2
2x^4-3x^3-6x^2+3x+2=0
两边同时除以x^2,得：2(x^2+1/x^2)-3(x-1/x)-6=0
代入t,得：2(t^2+2)-3t-6=0
2t^2-3t-2=0
(2t+1)(t-2)=0
t=-0.5,2
再将t=x-1/x化为2次方程：x^2-tx-1=0.即解得x.
t=-0.5时,x=(-0.5+√4.25)/2,(-0.5-√4.25)/2
t=2时,x=1+√2,1-√2
3x^4-4x^3-10x^2+4x+3=0
也一样,化为3(x^2+1/x^2)-4(x-1/x)-10=0
3(t^2+2)-4t-10=0
3t^2-4t-4=0
(3t+2)(t-2)=0
t=-2/3,2
t=-2/3时,x=-1/3+√10/3,-1/3-√10/3
t=2时,x=1+√2,1-√2

比如3/x^3这一项,分母趋向于无穷,那么这个极限就是0了

答：
(3x^4+2x^2)/(x^2+1)
=[3(x^2+1)x^2-(x^2+1)+1]/(x^2+1)
=3x^2-1+1/(x^2+1)
∫((3x^4+2x^2)/(x^2+1))dx
=∫(3x^2-1)dx+∫[1/(x^2+1)]dx
=x^3-x+arctanx+C