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
56
Actually none of the above
∫(2t^3 + 3) dt for t=-3 to 1
= [ (1/2)t^4 + 3t] from -3 to 1
= (1/2)(1) + 3 - ( (1/2)(81) -9)
= 1/2 - 81/2 + 12
= -40 + 12
= -28
To evaluate the given expression, we need to differentiate the definite integral.
First, let's evaluate the integral:
∫[1 to -3] (2t^3 + 3) dt
To integrate the polynomial function 2t^3 + 3, we can use the power rule of integration.
The power rule states that integrating t^n with respect to t gives us (1/(n+1)) * t^(n+1), where n is a constant.
Applying the power rule, we get:
∫(2t^3 + 3) dt = (2/4) * t^4 + (3 * t) + C
where C is the constant of integration.
Now, we evaluate the definite integral:
∫[1 to -3] (2t^3 + 3) dt = ((2/4) * (-3)^4 + (3 * (-3))) - ((2/4) * (1)^4 + (3 * (1)))
Simplifying the expression further:
= (2/4) * (-81) - 9 - (2/4) * 1 + 3
= (-162/4) - 9 - (1/2) + 3
= -40.5 - 9 - 0.5 + 3
= -47
Now, we differentiate the result with respect to x using the chain rule:
d/dx (-47) = 0
So, the answer is None of the above; the result is 0.