Green Function Homework: Problem & Solutions

[unscientific]

Homework Statement

The problem is attached in the first picture, the provided solutions are in the second.

The Attempt at a Solution

I got to where they are, but aren't they missing an additional term of sin(t)*cos(∏)*f(∏) from the second integral in dx/dt ?

 

[Millennial]

http://en.wikipedia.org/wiki/Differentiation_under_the_integral_sign

They aren't missing a term, read that.

 

[unscientific]

http://en.wikipedia.org/wiki/Differentiation_under_the_integral_sign

They aren't missing a term, read that.

Ok! I worked it out finally. Will the leibniz rule work if i take 'cos(t)' and 'sin(t)' out of the integral such that the inside of the integral only becomes one variable: ∫-sin(ζ)f(ζ) and ∫-cos(ζ)f(ζ) ??


The answer seems to use the leibniz rule but they imply that the 'cos(t)' and 'sin(t)' were taken out of the integral..

 

[Millennial]

cos(t) and sin(t) are treated as constants in the integral because the integral is with respect to zeta, and the integral itself is treated as a constant in differentiation because the derivative is taken with respect to t. The function inside inside the integral that involves t is differentiated as usual, and the integral is treated as a constant so it is left as-is.

 

[unscientific]

cos(t) and sin(t) are treated as constants in the integral because the integral is with respect to zeta, and the integral itself is treated as a constant in differentiation because the derivative is taken with respect to t. The function inside inside the integral that involves t is differentiated as usual, and the integral is treated as a constant so it is left as-is.

Sorry I'm not sure what you mean but here's the method that I used (attached in the picture)

In the picture its ∫ f(x,t) from u to v.

In the question of this thread, I am above to take 'sin(t)' and 'cos(t)' out of the integration so that the integration only has 1 variable.

My question is:

How do we still apply (or can we) leibniz's rule if this is the case? (Only 1 variable)

 

[Millennial]

Yes, you can, just take f(x,t)=f(t) (check the link I gave about differentiation under the integral sign.)