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
More Related Questions
- calc - also: integral of tan^(-1)y dy how is integration of parts used in that? ...
- integration by parts - s- integral s ln (2x+1)dx ? = ln(2x+1)x - s x d( ln (2x+1...
- 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 ...
- College Calc - Problem: y=sqrtx ^1/x. Find DY/DX. ln(f(x))=lnx^1/2x or lnx to ...
- calc - how do you start this problem: integral of xe^(-2x) There are two ways: 1...
- Calculus - Match the rule with the title: ____ 3. d/dx [f(x)/g(x) ]=(g(x) f^' (x...
- math - Use the Trapezoidal Rule, the Midpoint Rule, and Simpson's Rule to ...