Solved integral of the form ∫ln(αx+√(β+α^2x^2))dx

Solved integral of the form ∫ln(αx+√(β+α^2x^2))dx

Solving an indefinite integral of the form ∫ln(αx+√(β+α^2x^2))dx by the method of integration by parts, where α takes the values 1,2,3,4,5,…n; β takes values 1,2,3,4,5,…m..
An example of solving integrals for α=1, β=3 is considered; α=2, β=4; α=3, β=1 α=4, β=5.

∫ln(αx+√(β+α^2x^2))dx, ∫ln(x+√(3+x^2))dx, ∫ln(2x+√(4+4x^2))dx, ∫ln( 3x+√(1+9x^2))dx, ∫ln(4x+√(5+16x^2))dx

The solution is in PDF format