Evaluate using trig substitution?

Question

integral x/sqrt(9+x^2) dx

Answer 1

The integral expression is ʃ [x/√(9+x²)] dx.

Let x = 3 tan(θ) ⇒ dx = 3 sec²(θ) dθ and √(9+x²) = 3 sec(θ).

ʃ [x/√(9+x²)] dx

= ʃ 3 tan(θ) 3 sec²(θ) dθ /3 sec(θ)

= 3 ʃ [tan(θ) sec(θ)] dθ

= 3 sec(θ) + C

= 3 √{1+tan²(θ)} + C         [Since sec(θ) = √{1+tan²(θ)}]

= √{9+9tan²(θ)} + C          [Since 3 = √9]

= √(9+x²) + C                     [Since x = 3 tan(θ)]