求不定积分,分子是1,分母是x^1/2+x^1/4

题目

求不定积分,分子是1,分母是x^1/2+x^1/4

答案

设x^1/4=t
x^1/2=t^2
x=t^4
dx=4t^3dt
∫dx/(x^1/2+x^1/4 )= ∫4t^3dt/(t^2+t)
= ∫4t^2dt/(t+1)
=4∫(t^2-1+1)dt/(t+1)
=4∫(t-1)dt+4∫dt/(t+1)
=4*(t^2/2-t)+4ln|t+1|+C
=2t^2-4t+4ln(t+1)+C
=2x^1/2-x^1/4+4ln(x^1/4+1)+C