Answer to Question #5231 in Calculus for Jessica

Question

f(x)=fourth root of x^3 + cubed root of x^4. Find the most general antiderivative of the function. (Check your answer by differentiation)

Answer 1

Let f(x)=fourth root of x^3 + cubed root of x^4
= root[4]{x^3} +
root[3]{x^4} =
= x^{3/4} + x^{4/3}

Find the most general
antiderivative of the function.
(Check your answer by
differentiation)

Solution.
The antiderivative of f is

F(x) =
integral f(x) dx =
= integral ( root[4]{x^3} + root[3]{x^4} ) dx =

= integral ( x^{3/4} + x^{4/3} ) dx =
= x^{3/4+1} / (3/4 + 1)
+ x^{4/3+1} / (4/3 + 1) + C
= 4 x^{7/4} / 7 + 3 x^{7/3} / 7 +
C

Let us verify that that F'(x) = f(x).

F'(x) = (7 x^{7/4} / 4 +
7 x^{7/3} / 3 + C )' =
= 4/7 * 7/4 x^{7/4-1} + 3/7 * 7/3 x^{7/3-1}
=
= x^{3/4} + x^{4/3} =
= f(x).