# What is the antiderivative of F(x)=9x^2-8x+8?

Jul 15, 2015

$3 {x}^{3} - 4 {x}^{2} + 8 x + C$

#### Explanation:

The integral $\int {x}^{n} \setminus \mathrm{dx} = \frac{{x}^{n + 1}}{n + 1} + C$ when $n \ne - 1$ should be memorized (it's the reverse of the Power Rule $\frac{d}{\mathrm{dx}} \left({x}^{n}\right) = n {x}^{n - 1}$).

Also, up to addition of a constant of integration $C$, indefinite integrals satisfy the linearity property:

$\int \left({a}_{1} {f}_{1} \left(x\right) + {a}_{2} {f}_{2} \left(x\right) + \setminus \cdots + {a}_{n} {f}_{n} \left(x\right)\right) \setminus \mathrm{dx}$

$= {a}_{1} \int {f}_{1} \left(x\right) \setminus \mathrm{dx} + {a}_{2} \int {f}_{2} \left(x\right) \setminus \mathrm{dx} + \setminus \cdots + {a}_{n} \int {f}_{n} \left(x\right) \setminus \mathrm{dx}$

This means that we integrate term-by-term and "carry the constants along for the ride", so to speak. Then, tack on the $+ C$ at the end.