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不定积分∫1/1+t^3dt怎么求?
题目详情
不定积分∫1/1+t^3dt怎么求?
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答案和解析
∫1/(1+t^3)dt=∫1/[(1+t)(1-t+t^2)]dt=1/3∫1/[(1+t)dt-1/3∫(t-2)/(1-t+t^2)]dt=ln|t+1|/3-1/3∫(t-2)/[(t-1/2)^2+3/4]dt
=ln|t+1|/3-1/3∫(t-1/2)/[(t-1/2)^2+3/4]dt+1/2∫1/[(t-1/2)^2+3/4]dt
=ln|t+1|/3-1/6ln(t^2-t+1)+1/2arctan(t-1/2)+c
=ln|t+1|/3-1/3∫(t-1/2)/[(t-1/2)^2+3/4]dt+1/2∫1/[(t-1/2)^2+3/4]dt
=ln|t+1|/3-1/6ln(t^2-t+1)+1/2arctan(t-1/2)+c
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