Question

How do you find the antiderivative of `f(x)=-1/2x^3+2x^2-3x-2`?

Answer 1

`-1/8x^4+2/3x^3-3/2x^2-2x+C`

Explanation:

The antiderivative of the function is basically the integral of the function. So here we have: `int-1/2x^3+2x^2-3x-2 \ dx`.

We use the power rule and the constant rule, which states that,

• `inta^n \ dx=(a^(n+1))/(n+1)+C`

• `inta*f(x) \ dx=aintf(x) \ dx`,

respectively.

`=-1/8x^4+2/3x^3-3/2x^2-2x+C`

Answer 2

`-1/8x^4+2/3x^3-3/2x^2-2x+c`

Explanation:

`"integrate each term using the "color(blue)"power rule"`

`•color(white)(x)int(ax^n)=a/(n+1)x^(n+1)ton!=-1`

`int(-1/2x^3+2x^2-3x-2)dx`

`=-1/8x^4+2/3x^3-3/2x^2-2x+c`

`"where c is the constant of integration"`