Like differentiation, antidifferentiation is a linear operator that "goes past" sums and multiplicative constants. Applying the Power rule (Table 3.10.1) to each power of $x$ in the expression for $f\left(x\right)\=3 x^2-6 x^{1\/2}\+5$, the general antiderivative is

$F\left(x\right)\=3 \frac{x^3}{3}-6 \frac{x^{3\/2}}{3\/2}\+5 x\+c\=x^3-4 x^{3\/2}\+5 x\+c$ 

Solve the equation $F\left(4\right)\=64-32\+20\+c\=15$ for $c\=-37$ so that the desired antiderivative is

$F\left(x\right)\=x^3-4 x^{3\/2}\+5 x-37$ 

The following solution is obtained with the AntiderivativePlot command.

Note the value option used in the AntiderivativePlot command. The list of two numbers $\left[\mathrm{\α}\,\mathrm{\β}\right]$ assigned to this option determine $c$ in the equation $F\left(\mathrm{\α}\right)\=\mathrm{\β}$.

To obtain a solution from first principles, obtain the general antiderivative with the rules in Table 3.10.1 or with the tool in Table 3.10.1(a). Then solve the equation $F\left(4\right)\=15$ for the additive constant $c$ as in the Mathematical Solution above.

Extract the antiderivative from Table 3.10.1(a) via copy/paste.