Expert: Paul Klarreich - 11/22/2009

Question
Dear Paul, please help me with the following problems. Thank you so much.
a. Use integration by parts to find ∫(x^n)ln(ax)dx  (n not equal to -1)

b. Find the length of the curve y=x^(2/3)between x=-1 and x=8.

c.Use the method of Lagrange-multipliers to find the stationary values of z where z=x(y+4)    s.t x+y=8

In above problem (c), use the bordered Hessian to determine whether the stationary value of z is a maximum or a minimum.

Answer
Questioner: Geli
Country: United States
Category: Advanced Math
Private: No
Subject: Integral, maximum or minimum
Question: Dear Paul, please help me with the following problems. Thank you so much.
a. Use integration by parts to find ∫(x^n)ln(ax)dx  (n not equal to -1)

b. Find the length of the curve y=x^(2/3)between x=-1 and x=8.

c.Use the method of Lagrange-multipliers to find the stationary values of z where z=x(y+4)    s.t x+y=8

In above problem (c), use the bordered Hessian to determine whether the stationary value of z is a maximum or a minimum
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Hi, Geli,

When you send me a bunch of problems, I will just get you started -- this is not a do-your-homework-for-you site.


a. Use integration by parts to find ∫(x^n)ln(ax)dx  (n not equal to -1)

I think  u = ln(ax),  dv = x^n dx   should do it.

b. ds = sqrt(1 + (dy/dx)^2) dx

and dy/dx = 2/3 x^-1/3

So ds = sqrt(1 + 4/9 x^-2/3)

use the INTEGRATOR to get a hint on integrating it.


c.Use the method of Lagrange-multipliers to find the stationary values of z where z=x(y+4)    s.t x+y=8

Write  G(x,y) = x(y + 4) + lambda(x + y - 8)

Gx = y + 4 + lambda
Gy = x  + lambda
Glambda = x + y - 8

Set those to zero.