Evaluate the following integrals : ∫x/√(x^2+6x+10)dx ​

Question

Evaluate the following integrals :

\(\int\frac{x}{\sqrt{x^2+6x+10}}\)dx

Answer 1

Given,

I = \(\int\frac{x}{\sqrt{x^2+6x+10}}\)dx

Integral is of form

\(\int\frac{px+q}{\sqrt{ax^2+bx+c}}\)dx

Writing numerator as,

px + q = λ \(\{\frac{d}{dx}(ax^2+bx+c)\}+μ \)

⇒ px + q = λ(2ax + b) + μ 

⇒ x = λ (2x + 6) + μ 

∴ λ = 1/2 and μ = -3 

Let x = 1/2(2x + 6) – 3 and split,