Question

What is `f(x) = int -x+4 dx` if `f(1)=-2`?

Answer 1

`-1/2x^2+4x-11/2`

Explanation:

Integrate each term using `color(blue)"power rule for integration"`

`color(orange)"Reminder " color(red)(bar(ul(|color(white)(a/a)color(black)(int(ax^n)dx=a/(n+1)x^(n+1))color(white)(a/a)|)))`

`rArrint(-x+4)dx=-1/2x^(1+1)+4x^(0+1)+c`

`rArrF(x)=-1/2x^2+4x+c`

where c is the constant of integration.

We can find the value of c, using f(1) = -2. Substitute in F(x)

That is `(-1/2xx1^2)+(4xx1)+c=-2`

`rArr-1/2+4+c=-2rArrc=-2-4+1/2=-11/2`

Thus `F(x)=-1/2x^2+4x-11/2`