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calculus- help with integrals?
im working on an assignment and im having trouble with following question:
find the indefinite integral F of the function f(t) = 4t^3 + 2√t -7/t^2. Find the antiderivative of f(t) that satisfies f(1)=1
thanks
1 Answer
- 1 decade agoFavorite Answer
First, get all t's to have a positive exponent:
f(t) = 4t^3 + 2√t -7/t^2
= 4t^3 + 2t^(1/2) - 7t^(-2)
Then antidifferentiate according to the general rule f(x) = (x^(n + 1))/(n + 1) + c
to give you:
F(t) = t^4 + (4/3)t^(3/2) + 7t^(-1) + c
= t^4 + (4/3)t^(3/2) + 7/t + c
Now substitute the point (1;1) and solve for c:
1 = 1^4 + (4/3)(1)^(3/2) + 7/1 + c
1 = 1 + 4/3 + 7 + c
c = -25/3
Finally rewrite you function with the value of c written in:
F(t) = t^4 + (4/3)t^(3/2) + 7/t - 25/3
