Algebraic substitution Vs Trigonometric substitution

Problem

Evaluate the integral:

Algebraic substitution

Let :

From (1),

Using logarithmic differentiation,

Trigonometric substitution

Put x = sin q , , dx = cos q dq.

Since x = sin q,

From (6),

Note

The solution by algebraic substitution as in (5),

and by trigonometric substitution as in (7)

seems not the same.

Reader interested may try to prove that the two solutions differ by a constant.