换元法求定积分的思维方法及常见错误剖析

针对换元法求不定积分和求定积分时经常会出现的错误,提出在求解时要注意换元的条件,要满足在积分区间上单调且具有连续导数.在作变量替换的同时,相应替换积分的上下限.被积函数f(x)、积分上下限a,b]、积分变元的微分dx三者要同时替换.换元后不必换成原定积分的变量,直接用牛顿—莱布尼兹公式计算.

Thinking method of solving definite integral by changing variable and the analysis of common mistakes

Changing variable is one of the important ways of solving invariant integral and definite integral.When one applies integration by substitution of definite integral,he should pay attention to the substitution condition.The integrating range should be humdrum and have continuous derivative.When the variable is substituted,substitute the upper and lower limit of integral accordingly.The integrand f(x),upper limit and lower limit and differential of integral variable dx should be substituted at the same time.A...