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Q17  Integrate the functions \frac{x+ 2 }{\sqrt { x ^2 -1 }}

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let x+2 =A\frac{d}{dx}(x^2-1)+B=A(2x)+B
By comparing the coefficients and constant term on both sides, we get;

A=1/2 and B=2
then  x+2 = \frac{1}{2}(2x)+2

\int \frac{x+2}{\sqrt{x^2-1}}dx =\int\frac{1/2(2x)+2}{x^2-1}dx
                                \\=\frac{1}{2}\int\frac{(2x)}{\sqrt{x^2-1}}dx+\int \frac{2}{\sqrt{x^2-1}}dx\\ =\frac{1}{2}[2\sqrt{x^2-1}]+2\log\left | x+\sqrt{x^2-1} \right |+C\\ =\sqrt{x^2-1}+2\log\left | x+\sqrt{x^2-1} \right |+C

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manish

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