Question

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.)

f '''(t) = (t)^1/2 − 9 cos(t)
f(t) = _______.

Answer 1

You just need to integrate 3 times:

`f'''(t)=t^(1/2)-9\cos t`

`f''(t)=\displaystyle\int f'''(t)\,\mathrm dt=\frac23 t^(3/2)-9\sin t+C`

`f'(t)=\displaystyle\int f''(t)\,\mathrm dt=\frac4{15} t^(5/2)+9\cos t+Ct+D`

`f(t)=\displaystyle\int f'(t)\,\mathrm dt=\frac8{105} t^(7/2)+9\sin t+\frac C2 t^2+Dt+E`