# Substitution/Integration by Parts

• Jan 31st 2008, 11:51 AM
strgrl
Substitution/Integration by Parts
Hi, I'm having a little trouble with a substitution/integration by parts question...

find the integral of: ln(sqrt(x) +1)

the book lists the first step as substituting y = sqrt(x) +1

from there i tried integration by parts using u=lny, v'=1 but it didn't quite work out

the solution in meant to be: sqrt(x) - x/2 + (x-1)ln(1+sqrt(x))

any suggestions?

thank you!
• Jan 31st 2008, 12:40 PM
Krizalid
Quote:

Originally Posted by strgrl
find the integral of: ln(sqrt(x) +1)

the book lists the first step as substituting y = sqrt(x) +1

Now contemplate its derivative $dy=\frac1{2\sqrt x}\,dx.$

This means that $dy = \frac{1}
{{2(y - 1)}}\,dx.$
So

$\int {\ln \left( {1 + \sqrt x } \right)\,dx} = 2\int {(y - 1)\ln y\,dy} .$

From here it's quite straightforward, so just split the original integral into two pieces, both of them are easy to tackle.

(If $x$ was bounded on $[0,1],$ a definite integral, I could kill this with a double integration trick.)
• Jan 31st 2008, 02:32 PM
strgrl
thank you!