Question

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

Answer 1

`3x^3-4x^2+8x+C`

Explanation:

The integral `int x^(n)\ dx=(x^(n+1))/(n+1)+C` when `n!=-1` should be memorized (it's the reverse of the Power Rule `d/dx(x^(n))=nx^(n-1)`).

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

`int (a_{1}f_{1}(x)+a_{2}f_{2}(x)+\cdots+a_{n}f_{n}(x))\ dx`

`=a_{1}int f_{1}(x)\ dx+a_{2}int f_{2}(x)\ dx+\cdots+a_{n}int f_{n}(x)\ 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.