Posted by **marsha** on Wednesday, May 23, 2007 at 5:29pm.

evaluate the integral:

y lny dy

i know it's integration by parts but i get confused once you have to do it the second time

Leibnitz rule (a.k.a. product rule):

d(fg) = f dg + g df

y lny dy = d[y^2/2 ln(y)] - y/2 dy

---->

Integral of y lny dy =

y^2/2 ln(y) - y^2/4 + const.

Instead of partial integration you can use this trick:

Integral of y^a dy = y^(a+1)/(a+1) (1)

Differentiate both sides w.r.t. a:

Integral of y^a Ln(y) dy =

y^(a+1)Ln(y)/(a+1) -y^(a+1)/(a+1)^2 (2)

uUp to an integration constant)

Substitute a = 1 to obtain the answer. Note that the integration constant we could have added to (1) can still depend on the parameter a. So, if you differentiate it you get an integration constant in (2)

## Answer This Question

## Related Questions

- calc - also: integral of tan^(-1)y dy how is integration of parts used in that? ...
- math: pre-calculus - Let u=lnx and v=lny. Write the expression ln3√x/2y in...
- Calculus - L'Hospital's Rule Lim(lnx)^(x-1) x->1+ This is what I did so far...
- integration by parts - s- integral s ln (2x+1)dx ? = ln(2x+1)x - s x d( ln (2x+1...
- calculus - evaluate the double integral ∫R∫ lny/x dA for the region ...
- math: pre-calculus - Let u=ln and v=ln y. Write the expression ln(5√(x3&#...
- brief calc - evaluate the integral using integration by parts 7xe^-x
- Calculus - Would someone clarify this for me... Is antiderivatives just another ...
- math - I heard that when we are proving Leibniz's formula for differentiating an...
- Calc- Derivatives - So, I have a bunch of derivative problems for homework and ...

More Related Questions