what is the antiderivative of f(x)= x^4-2 sq root(X) +3/x^3

To find the antiderivative of f(x) = x^4 - 2√(x) + 3/x^3, we need to apply the rules of antiderivatives or integrate each term separately.

1. The antiderivative of x^n with respect to x (where n is any real number except -1) is (1/(n+1)) * x^(n+1). Applying this rule to the term x^4, we have:
∫ x^4 dx = (1/(4+1)) * x^(4+1) = (1/5) * x^5

2. The antiderivative of √(x) (square root of x) with respect to x is (2/3) * x^(3/2). Applying this rule to the term -2√(x), we have:
∫ -2√(x) dx = -2 * (2/3) * x^(3/2) = -(4/3) * x^(3/2)

3. The antiderivative of 1/x^n with respect to x (where n is any real number except 1) is (1/(1-n)) * x^(1-n). Applying this rule to the term 3/x^3, we have:
∫ 3/x^3 dx = 3 * (1/(1-3)) * x^(1-3) = 3 * (-1/2) * x^(-2) = -3/2x^2

Combining these results, the antiderivative of f(x) = x^4 - 2√(x) + 3/x^3 is:
F(x) = (1/5) * x^5 - (4/3) * x^(3/2) - (3/2) * x^(-2)

Note: The "+ C" is not included in this answer, as it represents the constant of integration and can be added to the antiderivative.