Question

求不定积分,分子是1,分母是x^1/2+x^1/4

Answer 1

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