Im unable to simplify this, ive tried but i just cant seem to get it to work.. it goes a little like this -
I understand that
But i need to simplify this -
Which i think should eventually end up like this -
Can someone show me the steps please? Btw, i posted this in calculus, because it had origianlly started off as an integration question.
(1/256){(1/3)tan^3(arcsin(x/4)) +tan(arcsin(x/4))}
Angle arcsin(x/4):
opp side = x
hyp = 4
adj side = sqrt(16 -x^2)
so,
tan(arcsin(x/4)) = opp/adj = x /sqrt(16 -x^2)
= (1/256){(1/3)[x /sqrt(16 -x^2)]^3 +x/sqrt(16 -x^4)}
= (1/256){[x^3 / 3(sqrt(16 -x^2)^3] +x/sqrt(16 -x^4)}
Combine the two fractions inside the curly brackets into one fraction only. The common denominator is 3(sqrt(16 -x^2))^3.
= (1/256){[x^3 +x*3(sqrt(16 -x^2))^2] / [3(sqrt(16 -x^2)^3]}
= (1/256){[x^3 +3x(16 -x^2)] / [3(sqrt(16 -x^2)^3]}
= (1/256){[x^3 +48x -3x^3] / [3(sqrt(16 -x^2)^3]}
= (1/256){[48x -2x^3] / [3(sqrt(16 -x^2)^3]}
= (1/256)(2/3){[24x -x^3] / [(sqrt(16 -x^2)^3]}
= (1/384){[x(24 -x^2)] / [(sqrt(16 -x^2)^3]}
or,
= x(24 -x^2) / 384(16 -x^2)^(3/2) -----------answer.