(x^2-3)/(5x-x^3)dx积分如何求,
来源:学生作业帮 编辑:作业帮 分类:数学作业 时间:2024/08/31 17:08:40
(x^2-3)/(5x-x^3)dx积分如何求,
∫(x^2-3)dx/(5x-x^3)
=∫(1/3)(3x^2-5)dx/(5x-x^3)+(-4/3)∫dx/x(5-x^2)
=(-1/3)ln|5x-3x^2|+ (-2/3)∫2xdx/[x^2 (5-x^2)]
=(-1/3)ln|5x-x^3|+(-2/3)∫dx^2/[x^2(5-x^2)]
=(-1/3)ln|5x-x^3|+(-2/15)[∫dx^2/x^2+∫dx^2/(5-x^2)]
=(-1/3)ln|5x-x^3|+(-2/15)ln|x^2|/|5-x^2|+C
再问: 不对饿,和答案不一样
再答: 答案请附上 应该是同解
再问: 答案是(-3/5)lnx-(1/5)ln(5-x^2)+c
再答: (-1/3)ln|5x-x^3|+(-2/15)ln|x^2|/|5-x^2| =(-1/3)lnx+(-1/3)ln(5-x^2) +(-4/15)lnx+(2/15)ln(5-x^2) =(-1/3-4/15)lnx+(-1/3+2/15)ln(5-x^2) =(-9/15)lnx+(-3/15)ln(5-x^2) =(-3/5)lnx+(-1/5)ln(5-x^2)
再问: 这个明白了,但是你上面求积分的第一步我没懂,什么方法饿?
再答: 拆分和凑微分 (5x-x^3)'=5-3x^2 x^2-3=(1/3)(3x^2-5)+(-4/3)
=∫(1/3)(3x^2-5)dx/(5x-x^3)+(-4/3)∫dx/x(5-x^2)
=(-1/3)ln|5x-3x^2|+ (-2/3)∫2xdx/[x^2 (5-x^2)]
=(-1/3)ln|5x-x^3|+(-2/3)∫dx^2/[x^2(5-x^2)]
=(-1/3)ln|5x-x^3|+(-2/15)[∫dx^2/x^2+∫dx^2/(5-x^2)]
=(-1/3)ln|5x-x^3|+(-2/15)ln|x^2|/|5-x^2|+C
再问: 不对饿,和答案不一样
再答: 答案请附上 应该是同解
再问: 答案是(-3/5)lnx-(1/5)ln(5-x^2)+c
再答: (-1/3)ln|5x-x^3|+(-2/15)ln|x^2|/|5-x^2| =(-1/3)lnx+(-1/3)ln(5-x^2) +(-4/15)lnx+(2/15)ln(5-x^2) =(-1/3-4/15)lnx+(-1/3+2/15)ln(5-x^2) =(-9/15)lnx+(-3/15)ln(5-x^2) =(-3/5)lnx+(-1/5)ln(5-x^2)
再问: 这个明白了,但是你上面求积分的第一步我没懂,什么方法饿?
再答: 拆分和凑微分 (5x-x^3)'=5-3x^2 x^2-3=(1/3)(3x^2-5)+(-4/3)