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Evaluate the integral of 1 / (a2 + x2)

Evaluate the following integral for a \neq 0:

    \[ \int \frac{dx}{a^2 + x^2}. \]


Since a \neq 0 we may write

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

Now, make the substitution u = \frac{x}{a} and du = \frac{dx}{a}. Then we have,

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

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