How to Simplify Integration by Parts for a Complex Expression?

[Dell]

₄$$\int$$⁹$$\frac{1-\sqrt{x}}{\sqrt{x}+2}$$

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$$\int$$1-$$\frac{2}{t+2}$$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.