int_0^3sqrt(x^(2)+9) \ dx
let x^2 = 9 sinh^2 z, x = 3 sinh z, dx/dz = 3 cosh z
implies int_(sinh^(-1) 0)^(sinh^(-1) 1) sqrt(9 sinh^2 z+9) \ 3 cosh z \ dz
implies 9int_(sinh^(-1) 0)^(sinh^(-1) 1) cosh^2 z \ dz
Using
cosh 2x =cosh ^2 x+ sinh ^{2} x=2 sinh ^2 x+1 = 2 cosh^2 x-1
implies 9/2 int_(sinh^(-1) 0)^(sinh^(-1) 1) cosh 2z + 1\ dz
9/2 [ 1/2sinh 2z + z ]_(sinh^(-1) 0)^(sinh^(-1) 1)
Using sinh 2x = 2 sinh x cosh x
= 9/2 [ sinh z cosh z + z ]_(sinh^(-1) 0)^(sinh^(-1) 1)
= 9/2 [ sinh z sqrt(1 + sinh^2 z) + z ]_(sinh^(-1) 0)^(sinh^(-1) 1)
= 9/2 ( 1 sqrt(1 + 1^2 ) + sinh^(-1) 1 )
= 9/2 ( sqrt(2) + sinh^(-1) 1 )