作业帮求正割的三次方乘以正切的二次方的不定积分,
来源：学生作业帮 编辑：作业帮 分类：数学作业 时间：2024/05/16 05:55:52
求正割的三次方乘以正切的二次方的不定积分,
有递推公式∫ secⁿx dx = (secⁿ⁻²xtanx)/(n - 1) + [(n - 2)/(n - 1)]∫ secⁿ⁻²x dx
∫ sec³xtan²x dx
= ∫ sec³x(sec²x - 1) dx
= ∫ sec⁵x dx - ∫ sec³x dx
= (1/4)sec³xtanx + (3/4)∫ sec³x dx - ∫ sec³x dx
= (1/4)sec³xtanx - (1/4)[(1/2)secxtanx + (1/2)∫ secx dx]
= (1/4)sec³xtanx - (1/8)secxtanx - (1/8)ln|secx + tanx| + C
∫ sec³xtan²x dx
= ∫ sec³x(sec²x - 1) dx
= ∫ sec⁵x dx - ∫ sec³x dx
= (1/4)sec³xtanx + (3/4)∫ sec³x dx - ∫ sec³x dx
= (1/4)sec³xtanx - (1/4)[(1/2)secxtanx + (1/2)∫ secx dx]
= (1/4)sec³xtanx - (1/8)secxtanx - (1/8)ln|secx + tanx| + C