大师作文网根号（x2—1）的不定积分
来源：学生作业帮 编辑：大师作文网作业帮 分类：数学作业 时间：2024/11/13 17:16:38
根号（x2—1）的不定积分
∫√（x^2-1）dx =∫tanx * secx*tanxdx (第二类换元法：x=sect,t属于<0,π/2))
=∫sect(sect*sect-1)dt=∫sect*sect*sectdt-∫sectdt=∫sectdtant-∫sectdt
=secttant-∫tant*tant*sectdt-∫sectdt
即∫√（x^2-1）dx =∫tant * sect*tantdt= secttant-∫tant*tant*sectdt-∫sectdt
将等式右边的∫tant*tant*sectdt移到左边:
∫√（x^2-1）dx =∫tant * sect*tantdt=1/2 secttant-1/2∫sectdt
=1/2 secttant-1/2ln⁄sect+tant⁄+c=1/2x√(x^2-1)-1/2ln(x+√(x^2-1))+c
=∫sect(sect*sect-1)dt=∫sect*sect*sectdt-∫sectdt=∫sectdtant-∫sectdt
=secttant-∫tant*tant*sectdt-∫sectdt
即∫√（x^2-1）dx =∫tant * sect*tantdt= secttant-∫tant*tant*sectdt-∫sectdt
将等式右边的∫tant*tant*sectdt移到左边:
∫√（x^2-1）dx =∫tant * sect*tantdt=1/2 secttant-1/2∫sectdt
=1/2 secttant-1/2ln⁄sect+tant⁄+c=1/2x√(x^2-1)-1/2ln(x+√(x^2-1))+c