Find the antiderivative... ?

Question

Ok, so the question says:

Find the antiderivative F of f that satisfies the given condition.

f(x)=4-3(1+x^2)^-1 , F(1)=0

and

f(x)=4throot(x^3) + cuberoot(x^4) (find the most general antideriviate.

Thanks

Answer 1

f ' (x) = F(x) = 4x -3tan^-1(x) + C

F(1) = 0 = 4(1) -3tan^-1(1) + C

=> 0 = 4 - 3 (pie/4) + C

=> C = 3pie/4 - 4

F(x) = 4x -3tan^-1(x) + 3pie/4 - 4

4throot(x^3) = (x^3)^(1/4) = x^(3/4)  use power rule to take anti-derivative

(4/7)x^(7/4)

cuberoot(x^4) = (x^4)^(1/3) = x^(4/3)

(3/7)x^(7/3)