Trig sub integral problem

Question

Would anyone be able to help me through this problem using trig sub?

1. ∫ dx/ sqrt(4x^2 + 9)

Answer 1

Hi

∫ dx/ sqrt(4x^2 + 9)

First, you need to know ∫ du/√ u²+ a² = Ln ( u + √ u²+ a²) (I)

Second, 4x^2 + 9 = 4 ( x² + 9/4) then √4x^2 + 9 = 2 √x² + 9/4

Third, ∫ dx/ sqrt(4x^2 + 9) = 1/2 ∫ dx / √ (x² + 9/4)

Now, u = x and a = 3/4 base of (I) the answer of integral will be:

1/2 ∫ dx / √ (x² + 9/4) = 1/2 Ln ( x + √(x² + 9/4) ) + c

I hope it is useful,

Minoo