Evaluate the following expression:

d/dx (integration sign: upper=1 and lower= -3) (2t^3 + 3)dt =
I am given the following options:
2t^3 + 3
56
5
-28.0
None of the above
Would the result of the expression be 0 or 5???

To evaluate the given expression, which is d/dx (integration sign: upper=1 and lower= -3) (2t^3 + 3)dt, we need to find the derivative of the integral with respect to x.

In this case, since t is the variable of integration, we can treat it as a constant when taking the derivative with respect to x. Therefore, we can rewrite the expression as follows:

d/dx (integration sign: upper=1 and lower= -3) (2t^3 + 3)dt = integration sign: upper=1 and lower= -3 d/dx (2t^3 + 3)dt

Now, let's compute the derivative of 2t^3 + 3 with respect to x. Since t is treated as a constant, we have:

d/dx (2t^3 + 3) = 0

Therefore, the result of the given expression is 0.

The correct answer is None of the above.

To evaluate the given expression, we need to compute the derivative of the integral of the function (2t^3 + 3) with respect to x. However, we have been provided with the options, so we can determine the answer by evaluating each option.

Let's calculate the integral of the function first:
∫(2t^3 + 3) dt

To integrate the term 2t^3, we use the power rule of integration:
∫(2t^3) dt = (2/4)t^4 = (1/2)t^4

To integrate the constant term 3, we treat it as 3t^0:
∫3 dt = 3t

Now, the integral of the function is given by:
∫(2t^3 + 3) dt = (1/2)t^4 + 3t

Next, we need to take the derivative of this integral with respect to x (d/dx):
d/dx [(1/2)t^4 + 3t]

The derivative of (1/2)t^4 with respect to x is 0 since t does not depend on x.

The derivative of 3t with respect to x is 3(dt/dx) = 3(Δt/Δx).

Since we have not been given any relation between x and t, we cannot determine the value of Δt/Δx, and thus cannot evaluate the expression. Therefore, the correct answer is "None of the above."