(35S) Quick integration

[Albert Chan]

(11-19-2022 01:42 AM)Liamtoh Resu Wrote:  Can someone walk me through the symbolic calculation of the function?

∫(sqrt(1+4*x^2) dx)            // let y=2x, dy = 2 dx
= ∫(sqrt(1+y^2) (dy/2))       // let y=sinh(z), dy = cosh(z) dz
= ∫(cosh(z) (cosh(z)/2 dz))
= ∫((1+cosh(2z))/4 dz)         // cosh(z)^2 = (1+cosh(2z))/2
= (z + sinh(2z)/2)/4             // sinh(2z) = 2*sinh(z)*cosh(z)
= (z + sinh(z)*cosh(z))/4

∫(x = 0 .. 1)
= ∫(y = 0 .. 2)
= ∫(z = 0 .. asinh(2))
= (asinh(2) + 2*sqrt(5))/4
≈ 1.47894285754