主要内容:
通过根式换元、分项凑分以及分部积分法等相关知识,介绍不定积分∫x√(x+1)dx的三种计算方法和步骤。
根式换元法:
设√(x+1)=t,则x=(t^2-1),代入得:
∫x√(x+1)dx
=∫t*(t^2-1)d(t^2-1),
=2∫t^2*(t^2-1)dt,
=2∫(t^4-1t^2)dt,
=2/5*t^5-2/3*t^3+C,
=2/5*(x+1)^(5/2)-2/3*(x+1)^(3/2)+C,
根式部分凑分法
∫x√(x+1)dx
=∫x√(x+1)d(x+1),
=2/3∫xd(x+1)^(3/2),
=2/3*x(x+1)^(3/2)- 2/3∫(x+1)^(3/2)dx,
=2/3*x(x+1)^(3/2)- 2/3∫(x+1)^(3/2)d(x+1),
=2/3*x(x+1)^(3/2)- 4/15*(x+1)^(5/2)+C,
整式部分凑分法
A=∫x√(x+1)dx,
=(1/2)∫√(x+1)dx^2,
=(1/2)x^2√(x+1)-(1/2)∫x^2d√(x+1),
=(1/2)x^2√(x+1)-(1/4)∫x^2/√(x+1)dx,
=(1/2)x^2√(x+1)-(1/4)∫[x(x+1)-(x+1)+1]/√(x+1)dx,
=(1/2)x^2√(x+1)-(1/4)A+1/4∫√(x+1)dx-1/4∫dx/√(x+1),
(5/4)A=(1/2)x^2√(x+1)+1/4∫√(x+1)dx-1/2∫dx/2√(x+1),
A=(2/5)x^2√(x+1)+1/5∫√(x+1)d(x+1)-2/5√(x+1),
A=(2/5)x^2√(x+1)+2/15(x+1)^(3/2)-2/5*√(x+1)+C
可见,三种方法的最终结果的函数表现形式不一样,但最终均可化简成同一解析式。
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