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Evaluate the integral using substitution

by RoRi

Use the method of substitution to evaluate the following integral:

    \[ \int \frac{x \, dx}{\sqrt{1+x^2 + \sqrt{(1+x^2)^3}}}. \]

First, let’s simplify the integral some

    \begin{align*} \int \frac{x \, dx}{\sqrt{1+x^2 + \sqrt{(1+x^2)^3}}} &= \int \frac{x \, dx}{\sqrt{(1+x^2)(1 + \sqrt{1+x^2})}} \\[9pt] &= \int \frac{x \, dx}{\left(\sqrt{1+x^2} \right) \left( \sqrt{1+\sqrt{1+x^2}} \right)}. \end{align*}

Now, let

    \[ u = 1 + \sqrt{1+x^2} \qquad \implies \qquad du = \frac{x \, dx}{\sqrt{1+x^2}}. \]

Then, we can evaluate the integral

    \begin{align*} \int \frac{x \, dx}{\left( \sqrt{1+x^2} \right) \left( \sqrt{1+ \sqrt{1+x^2}} \right)} &= \int u^{-\frac{1}{2}} \, du \\ &= 2 u^{\frac{1}{2}} + C \\ &= 2 \sqrt{1+\sqrt{1+x^2}} + C. \end{align*}

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