[Note that d(tan θ)/dθ equals sec2θ; d(sec θ)/dθ equals tan θsec θ.]
Create the right triangle with angle θ having adjacent (horizontal) side of 8 and opposite (vertical) side of x; this gives the side opposite the right angle or hypotenuse as √(x2 + 82). Then tan θ = x/8.
Out of x = 8tan θ comes dx = 8sec2θ dθ. Write ∫[x3dx/√(x2 + 82)] as ∫[83tan3θ(8sec2θ)dθ/√(64tan2θ+64)].
Rewrite as ∫[84tan3θ(sec2θ)dθ/8√(tan2θ+1)] or 83∫[tan3θ(sec2θ)dθ/√(sec2θ)] or 83∫tan3θsecθdθ.
Write 83∫tan3θsecθdθ as 83∫(sec2θ−1)tanθsecθdθ which integrates to 83[(1/3)sec3θ−sec θ] or 83[(1/3)sec3(arctan (x/8))−sec (arctan (x/8))].
Since sec (arctan (x/8)) equals √(x2 + 82)/8, 83[(1/3)sec3(arctan (x/8))−sec (arctan (x/8))] is rewritten as
83[(1/3)(√(x2 + 82)/8)3 − √(x2 + 82)/8] which reduces to (x2 + 82)1.5/3 − 64(x2 + 82)0.5 + C.
