Find the specific antiderivative using the given information.

Question

f'(t) = 2t^-1 − 1/t⁴ and f(−1) = 0

Answer 1

f'(t) = 2t^-1 - t^-4

antiderivative = integral = f(t) = 2lnltl +1/3t^3 + c

f(-1) = 0 = 2lnl-1l + 1/3(-1)^3 + c = 0 - 1/3

c= -1/3

specific antiderivative = f(t) = 2lnltl + 1/3t^3 - 1/3

raise the power by 1, and divide by that same number

-4+1 =-3. That's the new exponent and divisor

integral of t^-4 = t^-3/-3= -1/3t^3

for 2t^-1 if you try to raise the power by 1 you get t^0 and divide by zero, but that's not allowed.

integral of dt/t = lnltl + c where ltl is the absolute value of t

you could have 2lnltl + c + 1/3t^3 + another constant, but that's unnecessary

Answer 2

For all n ∫tⁿdt = t^n+t/(n+1) + C