Help With Leibniz Rule (With X as the bounds and inside the function)

[Amoren]

Hey, need some help with a calculus homework assignment due this afternoon. The lecturer gave the formula for the Leibniz rule (specifically, d/dx[ integral from u(x) to v(x) of the function f(x,t) dt] = f(v)*dv/dx - f(u)*du/dx + integral from u to v of the partial_f/partial_t dt ), but didn't go into any examples of the process, and I unfortunately missed the tutorial that week, and it appears that no helper in the study hall I went too have so far been taught math of this level yet, and I'm hoping not to bother the actual teacher if I can help it.

I'm hoping to get some examples/further working in the rule in practice, so I can apply it in new equations, specifically towards the three I have for my homework assignment. So far the internet has proven quite disheartening in this regard, as while a quick search nets the equation, none seem to offer actual examples of this particular case that I need.

These are the three homework problems I need to solve with the Leibniz rule, and while I don't need the exact answers for these three, going through similar problems would be much appreciated;

a) d/dx [integral from x to x^2 | 1/ln(x + u) du]
b) d/dx [integral from 1/x to 2/x | (sin xt)/t dt]
c) d/dx [integral from 1 to 1/x | (e^(xt))/t dt]

 

[Amoren]

I 'think' I've gotten a hold of things now (managed to track down some tutorial notes). Assuming I'm correct, d/dx for question a should be; 2x/ln(x+x^2) - 1/(ln(2x) + integrand(x to x^2) of du/((u+x)(ln^2(u+x))).

 

[Ishuda]

I 'think' I've gotten a hold of things now (managed to track down some tutorial notes). Assuming I'm correct, d/dx for question a should be; 2x/ln(x+x^2) - 1/(ln(2x) + integrand(x to x^2) of du/((u+x)(ln^2(u+x))).

If I'm reading it correctly then yes, you have the answer for a.

 

[Amoren]

Alright, cool. As a double check, I got these values for b and c.

b) -sin(2)/x + sin(1)/x + integrate(1/x to 2/x) cos(xt) dt.

c) -e/x + integrate(1 to 1/x) ((x-1)*e^(xt))/t^2 dt.


Also, as an aside, does anyone know of a nicer way to integrate 2*sqrt(4 - x^2 - y^2) than the big, nasty thing Wolfram alpha spits out? Trying to find the volume of a cylinder of radius one bound within the space of a sphere of radius 2, which should be a double integral with the integrand set to my function for z, if I'm correct. However, setting the equation for the sphere, x^2 + y^2 + z^2 = 4, equal to z returns that big, nasty integrand which is not easy to work with. It's just a bonus credit question, but its relatively easy aside from that one point.