Let's find the antiderivative [tex]\( F(x) \)[/tex] of the given function [tex]\( f(x) = x^3 - 3x^{-4} + 4 \)[/tex], and ensure it satisfies the condition [tex]\( F(1) = 1 \)[/tex].
Step 1: Integrate the function [tex]\( f(x) \)[/tex] to find the indefinite integral [tex]\( F(x) \)[/tex]:
[tex]\[ \int f(x) \, dx = \int (x^3 - 3x^{-4} + 4) \, dx \][/tex]
To integrate term by term:
1. The antiderivative of [tex]\( x^3 \)[/tex] is:
[tex]\[ \int x^3 \, dx = \frac{x^4}{4} \][/tex]
2. The antiderivative of [tex]\( -3x^{-4} \)[/tex] is:
[tex]\[ \int -3x^{-4} \, dx = -3 \int x^{-4} \, dx = -3 \left( -\frac{1}{3} x^{-3} \right) = x^{-3} \][/tex]
3. The antiderivative of [tex]\( 4 \)[/tex] is:
[tex]\[ \int 4 \, dx = 4x \][/tex]
Combining these results, we get the general antiderivative as:
[tex]\[ F(x) = \frac{x^4}{4} + 4x + x^{-3} + C \][/tex]
where [tex]\( C \)[/tex] is the constant of integration.
Step 2: Use the initial condition [tex]\( F(1) = 1 \)[/tex] to determine the constant [tex]\( C \)[/tex]:
Substitute [tex]\( x = 1 \)[/tex] and [tex]\( F(1) = 1 \)[/tex] into the general antiderivative:
[tex]\[ 1 = \frac{1^4}{4} + 4(1) + 1^{-3} + C \][/tex]
Simplify the equation:
[tex]\[ 1 = \frac{1}{4} + 4 + 1 + C \][/tex]
[tex]\[ 1 = 5.25 + C \][/tex]
[tex]\[ C = 1 - 5.25 \][/tex]
[tex]\[ C = -4.25 \][/tex]
[tex]\[ C = -\frac{17}{4} \][/tex]
Step 3: Write the complete antiderivative with the determined constant [tex]\( C \)[/tex]:
[tex]\[ F(x) = \frac{x^4}{4} + 4x + x^{-3} - \frac{17}{4} \][/tex]
So, the antiderivative [tex]\( F(x) \)[/tex] that satisfies [tex]\( F(1) = 1 \)[/tex] is:
[tex]\[ F(x) = \frac{x^4}{4} + 4x + x^{-3} - \frac{17}{4} \][/tex]
