Integration by parts etc

Latkan · #1 $Old$ April 9th, 2009, 06:08 PM

Hey guys i was hoping you could help me with these intergration questions, these are from a past paper but there is no answers to them so i am stuck lol (wince in mental and physical pain) please help

(a) Let f(x,y)= x^lny. Find df/dx and df/dy (the d is the squiggly d for partial)

(b) Verify that

f(x, y) = xy + x/y
satisfies the equation:

y d^2f + 2x d^f = 2x (again the d's here are the squiggly ones for partial)
dy^2 dxdy

(c) Let y= sinh^ -1 x.

Use dy = ( dx )^-1 to find dy
dx dy dx

Use intergration by parts to find

∫ sinh^-1 x

P.S Although i would appreciate any help, what i would really love is for the method not the answer because whilst the answer will be very helpful i need and i want to be able to do other questions like this on my own and with a worked example to help i think i can become self sufficient

Jhevon · #2 $Old$ April 9th, 2009, 10:10 PM

Quote:

Originally Posted by Latkan $View Post$

(a) Let f(x,y)= x^lny. Find df/dx and df/dy (the d is the squiggly d for partial)

to find $\frac {\partial f}{\partial x}$ , treat y as a constant, and differentiate. so you can use the power rule, $\frac d{dx}x^n = nx^{n - 1}$ , where $n$ is a constant

to find $\frac {\partial f}{\partial y}$ , treat x as a constant and differentiate. so you can use the rule, $\frac d{dx}a^u = u'a^u \ln a$ , where $a > 0$ is a constant

Quote:

(b) Verify that

f(x, y) = xy + x/y
satisfies the equation:

y d^2f + 2x d^f = 2x (again the d's here are the squiggly ones for partial)
dy^2 dxdy

in light of the above, that is, knowing what to hold constant and what to let vary, find $\frac {\partial ^2 f}{\partial y^2}$ and $\frac {\partial f}{\partial x}$ , plug them into the left side of the equation and simplify to get the right side

Quote:

(c) Let y= sinh^ -1 x.

Use dy = ( dx )^-1 to find dy
dx dy dx

use what they said, first find $\frac {dx}{dy}$ , and then take its reciprocal to find $\frac {dy}{dx}$ .

note that $y = \sinh^{-1}x \implies x = \sinh y$ [font=Verdana]

Quote:

Use intergration by parts to find ∫ sinh^-1 x

recall the integration by parts formula, $\int u~dv = uv - \int v~du$

Here, take $u = \sinh^{-1}x$ and $dv = dx$