求[1+X^(1/2)] / [ 2X+X^(3/2) ]的積分 謝謝了!!!?

Sol
∫[1+x^(1/2)] / [ 2x+x^(3/2) ]dx
Set y=x^(1/2)
y^2=x
dx=2ydy
∫[1+x^(1/2)] / [ 2x+x^(3/2) ]dx
=∫(1+y) /( 2y^2+y^3)*2ydy
=2∫(1+y) /( 2y+y^2)dy
設(2+2y)/(2y+y^2)=a/y+b/(y+2)
2+2y=a(y+2)+by
當y=0 時
2=2a
a=1
當y=－2 時
－2=b*(－2)
b=1
So
(2+2y)/(2y+y^2)=1/y+1/(y+2)
∫[1+x^(1/2)] / [ 2x+x^(3/2) ]dx
=∫1/yd+∫1/(y+2)dy
=ln｜y｜+ln｜y+2｜+c
=ln[x^(1/2)]+ln[x^(1/2)+2]+c