Find the anti-derivative F of f defined by f( | vedclass

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(A) The anti-derivative $F(x)$ is the integral of $f(x)$ with respect to $x$: $F(x) = \int (4x^{3} - 6) \, dx$ $F(x) = 4 \int x^{3} \, dx - 6 \int 1 \

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$x^{4} - 6x + 3$

Vedclass · Answer

(A) The anti-derivative $F(x)$ is the integral of $f(x)$ with respect to $x$: $F(x) = \int (4x^{3} - 6) \, dx$ $F(x) = 4 \int x^{3} \, dx - 6 \int 1 \, dx$ $F(x) = 4 \left( \frac{x^{4}}{4} \right) - 6x + C$ $F(x) = x^{4} - 6x + C$ Given that $F(0) = 3$,we substitute $x = 0$ into the equation: $3 = (0)^{4} - 6(0) + C$ $3 = 0 - 0 + C$ $C = 3$ Therefore,the required anti-derivative is $F(x) = x^{4} - 6x + 3$.

Find the anti-derivative $F$ of $f$ defined by $f(x) = 4x^{3} - 6$,where $F(0) = 3$.

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