Question

If (below) is f'(x) then what is f(x)? This is for both equations.

`sqrt(-(x+4)^2+1)`
`-sqrt(-(x+1)^2+4)`

Answer 1

`1/2(sin^(-1)(x+4)+(x+4)sqrt(1-(x+4)^2))` + C

Explanation:

`y'=sqrt(1-(x+4)^2)>=0`.

Upon substitution `x+4=sin theta`,

`y'=(dy)/(dx)=sqrt(1-(x+4)^2)=|cos theta|.`

So, `dy = |cos theta| dx`

`= |cos theta|(dx)/(d theta) d theta`

`=|cos theta|cos theta d theta=cos^2theta d theta`, as `y'>=0`..

Upon integration,

`y = int cos^2theta d theta`

`=1/2int (1+cos (2theta)) d theta`

`=1/2(theta +1/2sin (2theta))+C`

`=1/2(sin^(-1)(x+4)+sin theta cos theta)`+C

`=1/2(sin^(-1)(x+4)+(x+4)sqrt(1-(x+4)^2))` + C

I have paved the way towards the similar answer, for the other

function.