Question

What are they asking for here?

Answer 1

It says compute this derivative \[\frac{ d }{ dx }\int\limits_{0}^{x}(1+t^2)^4dt\]

Answer 2

Do i find the integral or do I find the derivative? I'm confused

Answer 3

Like why is there the 'd/dx' before the integral sign?

Answer 4

this should really be done by applying Leibnitz's rule for differentiating integrals.

however, it might be possible to take a short cut as it is a basic example.

imagine you could integrate this first, and you would get something like ∫ f(t) dt = F(x) - F(0) [where f(t) = (1+t^2)^4 and 0 and x are the limits] and F is the indef integral with the limits inserted.

so the integral is ultimately a function of x. and you would just then have to do d/dx {F(x) - F(0) } to get your answer.

however, its a pig of an integral.

but you can use the fact that dF/dt = f(t) [because F = ∫f(t) dt], and t is a function of x (of sorts!) as, in the limits, 0≤t≤x so you are subbing t = x and t - 0 into F.

and then, purely by chaining it, dF/dx = dF/dt * dt/dx = f(x) * 1

but as stated, if Leibnitz what you are learning, you might be better off plugging this into the whole Leibnitz shebang and getting a feel for the overall approach.

Answer 5

From Mathematica v9:\[\frac{\partial }{\partial x}\left(\int\limits_0^x \left(t^2+1\right)^4 \, dt\right)=x^8+4 x^6+6 x^4+4 x^2+1 \]