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Answer 1 · 2017-04-22T20:53:05

#8arcsin(x/4) + 4sin 2arcsin(x/4) + C#

#x=4sint => dx=4costdt#

#t = arcsin(x/4)#

#sqrt(16-x^2) = sqrt(16-(4sint)^2) = sqrt(16-16sin^2t) = #

#= sqrt(16(1-sin^2t)) = sqrt(16cos^2t) = 4cost#

#int sqrt(16-x^2)dx = int 4cost*4costdt = 16 int cos^2tdt = #

#16 int (1+cos2t)/2dt = 8[t+1/2sin2t]+C = #

#= 8arcsin(x/4) + 4sin 2arcsin(x/4) + C#