Question

Evaluate ∫ (2x+3)√(4x²+5x+6)dx

\(\int(2x+3)\sqrt{4x^2+5x+6}\) dx

Answer 1

∫ (2x+3)√(4x²+5x+6)dx

Make perfect square of quadratic equation,

4x² + 5x + 6

Let,

x + \(\frac{5}{8}\) = t

⇒ x = t - \(\frac{5}{8}\) 

Differentiate both side with respect to t,

\(\frac{dx}{dt}\) = t

⇒ dx = dt

Let, 

t² = z 

Differentiate both side with respect to z,

 \(2t\frac{dt}{dz}\) = 1

⇒ tdt = \(\frac{1}{2}\)dz