Fundamental Theorem of Calculus Help?

Question

Hello there, I've recently started learning about the fundamental theorem and have a few problems I'm not sure I understand how to do. Any help would be appreciated. 

1. Find F(x), if f(x) = ∫ (3t-t^2) dt (Range; top: x, bottom: 3)

2. Find F(x), if f(x) = ∫ sqrt(1-t) dt (Range; top: sin(x), bottom: -1)

Answer 1

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When you say "range," I guess you mean the limits on the integral.

#1. F(x) = (3/2)(x - 3) - (1/3)(x^3 - 27).

#2. F(x) = -(2/3)(1 - sin(x))^(3/2) + (2/3)(1 - (-1))^(3/2)

= -(2/3)(1 - sin(x))^(3/2) + (1/3)*2^(5/2).

Answer 2

1. f(x) = ∫ (3t-t^2) dt (Range; top: x, bottom: 3)

F(x) = t from 3 to x, ∫ (3t-t^2) dt = t from 3 to x,[(3t^2)/2 -(t^3)/3] 

F(x) = (3x^2)/2 - (x^3)/3 -27/2+27/3 =(3x^2)/2 - (x^3)/3 - 27/6. 

2. f(x) = t from -1 to sinx,∫ √(1-t) dt = t from -1 to sinx,(-2/3)[(1-t)^(3/2)].

F(x) = (-2/3)[(1-sinx)^(3/2) - 2^(3/2)].