Evaluate the following integrals : ∫2x+1/√(x^2+2x-1)dx

Question

Evaluate the following integrals :

\(\int\frac{2x+1}{\sqrt{x^2+2x-1}}\)dx

Answer 1

Given,

I = \(\int\frac{2x+1}{\sqrt{x^2+2x-1}}\)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

⇒ 2x + 1 = λ (2x + 2) + μ

⇒ x = λ (2x + 6) + μ 

∴ λ = 1 and μ = -1

Let 2x + 1 = 2x + 2 – 1 and split,