Find the antiderivative for each of the following, given an initial condition F(0) = k (a constant), or a boundary condition F

(a) = k (a ≠ 0).

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(a) [ (25x" (25x1/4 +16x1/3) dx, given the initial condition F(0) = 19. At F(0) = 19, c = 19, and (25x1/4 + 16x) dx = 25 5 x5/4 +16 4/3 +c =20x5/4 + 12x4/3 + c 19 = 20(0)5/4 + 12(0)4/3 + c F(x)=20x5/4+ 12x4/3+ 19 (b) (16e + 15e) dt, given F (0) = 9. (16 (16e2 + 15e-31) dt = 8e - 5e-31 + c L (16e2 At F(0) = 9, 9 = 8e2(0) - 5e -3(0) +c With e = 1, Thus, (c) (4x-1. 9=8-5+c c = 6 F(t)=8e25e-31 +6 +5x2) dx, given the boundary condition F(1) = 3. (4x-1. 5 = +5x2) dx 4 In |x| -5x+c=4 ln x x At F(1) = 3, 5 3 = 4 In 1 +c 1 3: c = 8 With In 1 = 0, and = -5+c F(x)=4lnx- 5 - - x +8 +c