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

Question

$\sqrt{- {(x + 4)}^{2} + 1}$ $sqrt(-(x+4)^2+1)$
$- \sqrt{- {(x + 1)}^{2} + 4}$ $-sqrt(-(x+1)^2+4)$

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Answer 1 · 2016-12-29T16:33:19

$\frac{1}{2} ({sin}^{- 1} (x + 4) + (x + 4) \sqrt{1 - {(x + 4)}^{2}})$ $1/2(sin^(-1)(x+4)+(x+4)sqrt(1-(x+4)^2))$ + C

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

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.