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

Question

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

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Answer 1 · 2016-12-29T16:33:19

#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.