Finding a particular antiderivative from an initial condition

Integrate term by term: the antiderivative of $8x$ is $4x^2$, and the antiderivative of $-3$ is $-3x$. So the general antiderivative is $f(x) = 4x^2 - 3x + C$, where $C$ is an arbitrary constant that cannot be omitted.

Now use the initial condition $f(1) = 5$: substitute $x=1$ to get $4(1)^2 - 3(1) + C = 5$, which simplifies to $1 + C = 5$, so $C = 4$. Therefore the particular antiderivative is $f(x) = 4x^2 - 3x + 4$.

The constant C was found by plugging the initial condition into the general antiderivative four x squared minus three x plus C. Substituting x equals one and setting the result equal to five gave one plus C equals five, so C equals four.