Definite integrals and one partial derivative.

[mmc5311]

Hey again.


1. ?x/(?x + 1) dx , from x = 0 to x = 3

I took ?x(x + 1)^-1/2


2. ?ln(x + 2) dx , from x = -1 to x = 2

With this one, I took the integral of ln(x + 2) and got 1/(x + 2). I would just start plugging in the x-values, but that seems to easy. Did I take the wrong first step?


For problems #1 and #2, I feel like I should be using differentiation by parts, but when I try, I always get confused. If that is the way to do it, please, there's no such thing as too much explanation when it comes to me.

3. Let f(x,y) = (x + y^2)^3. Find ?^2f/?x^2.

This one I couldn't even start on. I'm familiar with partial derivatives, but this problem is a little abnormal.
What are the steps I have to take to solve it?


Thanks.

 

[Deleted member 4993]

Hey again.


1. ?x/(?x + 1) dx , from x = 0 to x = 3

I took ?x(x + 1)^-1/2 <<<< I don't understand your solution.

Do integration by parts

u = x ? du = dx

dv = (x+1)[sup:4jdq02hm]-1/2[/sup:4jdq02hm] ? v = (x+1)[sup:4jdq02hm]1/2[/sup:4jdq02hm]

2. ?ln(x + 2) dx , from x = -1 to x = 2

Do integration by parts

u = ln(x+2) ? du = (x+2) [sup:4jdq02hm]-1[/sup:4jdq02hm] dx

dv = dx ? v = x

With this one, I took the integral of ln(x + 2) and got 1/(x + 2). <<< Incorrect

Derivative of ln(x+2) is (x+2) [sup:4jdq02hm]-1[/sup:4jdq02hm] - not the anti-derivative

I would just start plugging in the x-values, but that seems to easy. Did I take the wrong first step?


For problems #1 and #2, I feel like I should be using differentiation (integration) by parts, but when I try, I always get confused. If that is the way to do it, please, there's no such thing as too much explanation when it comes to me.

3. Let f(x,y) = (x + y^2)^3. Find ?^2f/?x^2.

first find ?f/?x - treating 'y' as constant

then find ?[sup:4jdq02hm]2[/sup:4jdq02hm]f/?x[sup:4jdq02hm]2[/sup:4jdq02hm] - treating 'y' as constant again

This one I couldn't even start on. I'm familiar with partial derivatives, but this problem is a little abnormal.
What are the steps I have to take to solve it?


Thanks.