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Evaluate the integral of 1 / (1-2x-x2)1/2

by RoRi

Evaluate the following integral:

    \[ \int \frac{dx}{\sqrt{1-2x-x^2}}. \]

First, we simplify the integrand

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

Now we make a substitution, letting u = \frac{x+1}{\sqrt{2}} which implies du = \frac{1}{\sqrt{2}} \, dx. Therefore,

    \begin{align*} \int \frac{dx}{\sqrt{1-2x-x^2}} &= \int \frac{du}{\sqrt{1-u^2}} \\ &= \arcsin u + C \\ &= \arcsin \frac{x+1}{\sqrt{2}} + C. \end{align*}

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