∫[1/(√4-9x^)]2dx
来源:百度知道 编辑:UC知道 时间:2024/05/23 14:59:05
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解:由于被积函数的定义域为[-2/3,2/3],所以设x=[2sin(t)]/3,t属于[0,pi/2)并(pi/2,pi];代入原式得:∫[1/(√4-9x^2)]dx
=∫[1/(2*cost)]dx
=(1/2)*∫[cost/(cost)^2]dx
=(1/2)*∫{cost/[(1-(sint)^2]}dt
=(1/4)*∫{[cost/(1+sint)]+[cost/(1-sint)]}dx
=(1/4)*{[∫d(sint)/(1+sint)]+[∫d(sint)/(1-sint)]}
=(1/4)*{[∫d(sint)/(1+sint)]-[∫d(sint)/(sint-1)]}
=(1/4)*ln[(1+sint)/(1-sint)]+C
=(1/4)*ln{(1+sint)^2/[(1-(sint)^2]}+C
=(1/4)*2*ln{[(1+sint)/cost]的绝对值}+C
=(1/2)*ln[(sect+tant)的绝对值]+C
=(1/2)*ln{(2+3x)/[根号下(4-9x^2)]}+C
注意:根据我们的换元可得sect+tant>0,所以2+3x不必加绝对值符号(实际上也不应该加).
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