How to Simplify Integration by Parts for a Complex Expression?

[Dell]

₄\int⁹\frac{1-\sqrt{x}}{\sqrt{x}+2}

what i did:
t=\sqrt{x}
x=t²
dx=x'dt=2tdt

now my integration is from 2-3 instead of 4-9

\int\frac{1-t}{t+2}2tdt=2\int\frac{t-t²}{t+2}dt

=2(\int\frac{t}{t+2}dt-\int\frac{t²}{t+2}dt)

the 1st integral i can replace with\int1-\frac{2}{t+2}dt which is an immediate integral, but how do i simplify the second part??

 

[Dick]

Just divide t+2 into t^2 using polynomial division. Actually, you might want to take a step by and divide t-t^2 by t+2.

 

[Dell]

i get t+3-[6/(t+2)]

so i get --->2[(t^2)/2]+3t-6ln|t+2|]
is this right??

 

[Dick]

i get t+3-[6/(t+2)]

so i get --->2[(t^2)/2]+3t-6ln|t+2|]
is this right??

Sure.