Question

question 2
find the most general antiderivative of the function.
$f(x)=15x^{2}-14x + 8$
$f(x)=6x^{2}-14x + c$
$f(x)=5x^{3}-7x^{2}+8x + c$
$f(x)=30x^{5}-28x^{4}+9x + c$
$f(x)=36x - 14 + c$
$f(x)=18x^{3}-14x^{2}+9x + c$

Explanation:

Step1: Apply the power rule for integration

The power rule for integration is \(\int x^{n}dx=\frac{x^{n + 1}}{n+1}+C\) (\(n
eq - 1\)).
For the function \(f(x)=15x^{2}-14x + 8\), we integrate each term separately.
For the term \(15x^{2}\): \(\int15x^{2}dx=15\times\frac{x^{2+1}}{2 + 1}=15\times\frac{x^{3}}{3}=5x^{3}\)
For the term \(-14x\): \(\int-14xdx=-14\times\frac{x^{1+1}}{1+1}=-14\times\frac{x^{2}}{2}=-7x^{2}\)
For the term \(8\): \(\int8dx=8x\) (since \(\int kdx=kx + C\) where \(k = 8\))

Step2: Combine the integrated terms

Combining the results of the integration of each term, we get \(F(x)=5x^{3}-7x^{2}+8x + C\)

Answer:

\(F(x)=5x^{3}-7x^{2}+8x + C\) (the second option)