Find f. (Use C for the constant of the first antiderivative, D for the constant of the second antiderivative and E for the constant of the third antiderivative.)

Question

f''''(t) = sqrt(t) - 6 cos (t), determine f(t)

Answer 1

I'm guessing that you are given the third derivative, not 4th, otherwise we run out of specified constants of integration. So if f'''(x) = t^1/2 - 6 cos t, then f''(x) = (2/3) t^3/2 - 6 sin t + C.

f'(x) = (2/3)(2/5) t^5/2 + 6 cos t + C t + D.

f(x) = (4/15)(2/7) t^7/2 + 6 sin t + (1/2)C t² + D t + E. First coefficient of course simplifies to 8/105.

If we are really starting from f'''', then integrate one more time and add an arbitrary letter like "F".