作业帮不定积分x^7dx/(x^4+2) 不定积分(3x^4+x^3+4x^2+1)dx/(x^5+2x^3+x) 不定积分dx/(x^4+x^2+1）
来源：学生作业帮助网 编辑：作业帮 时间：2024/04/20 19:07:56
不定积分x^7dx/(x^4+2) 不定积分(3x^4+x^3+4x^2+1)dx/(x^5+2x^3+x) 不定积分dx/(x^4+x^2+1）
不定积分x^7dx/(x^4+2) 不定积分(3x^4+x^3+4x^2+1)dx/(x^5+2x^3+x) 不定积分dx/(x^4+x^2+1）
不定积分x^7dx/(x^4+2) 不定积分(3x^4+x^3+4x^2+1)dx/(x^5+2x^3+x) 不定积分dx/(x^4+x^2+1）
∫x^7dx/(x^4+2)
=(1/4)∫x^4d(x^4)/(x^4+2)
=(1/4)x^4-(1/4)ln(x^4+2)+C
∫（3x^4+x^3+4x^2+1)dx/(x^5+2x^3+x)
=∫(3x^4+6x^2+3)dx/(x^5+2x^3+x)+∫(x^3-2x^2-2)dx/(x^5+2x^3+x)
=3∫dx/x+ ∫(x^3+x)dx/(x^5+2x^3+x)+∫-xdx/(x^5+2x^3+x) +∫(-2x^2-2)dx/(x^5+2x^3+x)
=3lnx+arctanx+∫-dx/(x^2+1)^2 +∫-2dx/x(x^2+1)
=3lnx+arctanx -∫dx/(x^2+1)^2+2∫(x^2-(x^2+1))dx/[x(x^2+1)]
=3lnx+arctanx-∫dx/(x^2+1)^2+2∫xdx/(x^2+1) -2∫dx/x
=lnx+arctanx-∫dx/(x^2+1)^2+ln(x^2+1)
∫dx/(x^2+1)^2 x=tanu ∫dx/(x^2+1)^2=∫secu^2du/secu^4=∫cosu^2du=u/2+sin2u/4
=arctanx /2+(1/2)x/ (1+x^2)
=lnx+arctanx/2 -(1/2)x/(1+x^2)+ln(x^2+1)+C
∫dx/(x^4+x^2+1)
=∫dx/[(x^2+x+1)(x^2-x+1)]
=(1/2)∫dx/[x(x^2-x+1)] -(1/2)∫dx/[x(x^2+x+1)]
∫dx/[x(x^2-x+1)] =∫[x^2-(x^2-x+1)]dx/[x(x-1)(x^2-x+1)
=∫xdx/[(x-1)(x^2-x+1)] - ∫dx/[x(x-1)]
=-∫(x-1)dx/(x^2-x+1)+∫dx/(x-1)-∫dx/(x-1)+∫dx/x
=(-1/2)ln(x^2-x+1)+(-1/2)∫dx/(x^2-x+1)+ln|x|
=(-1/2)ln(x^2-x+1)+(-1/(√3))arctan[(2x-1)/√3] +ln|x|+C
∫dx/[x(x^2+x+1)]=∫[(x^2+x+1)-x^2]dx/[x(x+1)(x^2+x+1)]
=∫dx/[x(x+1)]-∫xdx/[(x+1)(x^2+x+1)]
=ln|x|-ln|x+1| -∫[(x+1)^2-(x^2+x+1)]dx/[(x+1)(x^2+x+1)]
=ln|x|-ln|x+1|-∫(x+1)dx/(x^2+x+1)+∫dx/(x+1)
=ln|x|-(1/2)ln(x^2+x+1)-∫(1/2)dx/[(x+1/2)^2+3/4]
=ln|x|-(1/2)ln(x^2+x+1)-(1/√3)arctan[(2x+1)/√3]
代入
∫dx/(x^4+x^2+1)=(-1/4)ln(x^2-x+1)+(-1/(2√3))arctan[(2x-1)/√3]
+ (1/4)ln(x^2+x+1)+(1/(2√3))arctan[(2x+1)/√3]+C