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Q 21   Integrate the functions \frac{x +2 }{\sqrt { x^ 2 + 2x +3 }}

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\frac{x +2 }{\sqrt { x^ 2 + 2x +3 }}
\int \frac{x+2}{\sqrt{x^2+2x+3}}dx = \frac{1}{2}\int \frac{2(x+2)}{\sqrt{x^2+2x+3}}dx
                                           \\= \frac{1}{2}\int \frac{2x+2}{\sqrt{x^2+2x+3}}dx+\frac{1}{2}\int \frac{2}{\sqrt{x^2+2x+3}}dx\\ =\frac{1}{2}\int \frac{2x+2}{\sqrt{x^2+2x+3}}dx+\int \frac{1}{\sqrt{x^2+2x+3}}dx\\ I=\frac{1}{2}I_1+I_2...........(i)

take I_1

\int \frac{2x+2}{\sqrt{x^2+2x+3}}dx
let x^2+2x+3 = t \Rightarrow (2x+2)dx =dt

I_1=\int \frac{dt}{\sqrt{t}}=2\sqrt{t}=2\sqrt{x^2+2x+3}
considering I_2

= \int \frac{dx}{\sqrt{x^2+2x+3}}= \int \frac{dx}{\sqrt{(x+1)^2+(\sqrt{2})^2}}
                                           = \log \left | (x+1)+\sqrt{x^2+2x+3} \right |
putting the values  in equation (i)

I=\sqrt{x^2+2x+3} +\log \left | (x+1)+\sqrt{x^2+2x+3} \right |+C

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manish

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