2x-1-x-2-4x-5-dx-

Question Number 145775 by Engr_Jidda last updated on 08/Jul/21

\int \frac{2 x + 1}{\sqrt{x^{2} + 4 x + 5}} d x

Answered by mathmax by abdo last updated on 08/Jul/21

Υ = \int \frac{2 x + 4 - 3}{\sqrt{x^{2} + 4 x + 5}} dx = \int \frac{2 x + 4}{\sqrt{x^{2} + 4 x + 5}} dx - 3 \int \frac{dx}{\sqrt{x^{2} + 4 x + 5}}

= 2 \sqrt{x^{2} + 4 x + 5} - 3 I

I = \int \frac{dx}{\sqrt{x^{2} + 4 x + 4 + 1}} = \int \frac{dx}{\sqrt{{(x + 2)}^{2} + 1}}

=_{x + 2 = sht} \int \frac{cht}{cht} dt = t + K = argsh (x + 2) + K

= \log (x + 2 + \sqrt{1 + {(x + 2)}^{2}}) + K \Rightarrow

Υ = 2 \sqrt{x^{2} + 4 x + 5} - 3 \log (x + 2 + \sqrt{x^{2} + 4 x + 5}) + K

Answered by ArielVyny last updated on 08/Jul/21

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

t = x + 2 \to d t = d x

\int \frac{2 (t - 2) + 1}{\sqrt{t^{2} + 1}} d t = \int \frac{2 t}{\sqrt{t^{2} + 1}} d t - 3 \int \frac{1}{\sqrt{t^{2} + 1}} d t

2 \sqrt{t^{2} + 1} - 3 a r g s h (t) + c t e

Commented by ArielVyny last updated on 08/Jul/21

2 \sqrt{x^{2} + 2 x + 5} - 3 a r g s h (x + 2) + c t e