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Antiderivatives for Sums and Differences of Squares

Some formulas to compute antiderivatives of functions including sums or differences of squares are deduced.

In this unit some formulas to calculate the antiderivative of functions that include a^2 + u^2, a^2 - u^2 or u^2 - a^2 are deduced.

Antiderivative of y = \nicefrac{1}{(a^2 + u^2)}

Rewrite the denominator factoring a^2 from both terms as follows:

    \begin{equation*} 	\int\frac{du}{a^2 + u^2} = \int\frac{du}{\displaystyle a^2\left(1 + \frac{u^2}{a^2}\right)} \end{equation*}

Now, multiply in the denominator by 1, written as \nicefrac{a}{a}:

    \begin{equation*} 	\int\frac{du}{a^2 + u^2} = \frac{1}{a^2}\int\frac{\displaystyle\frac{a}{1}\cdot\frac{du}{a}}{\displaystyle 1 + \frac{u^2}{a^2}} 		= \frac{1}{a}\int\frac{\displaystyle\frac{du}{a}}{\displaystyle 1 + \frac{u^2}{a^2}} \end{equation*}

Define v =  \nicefrac{u}{a}, so that with the change of variable the expression becomes:

    \begin{equation*} 	\int\frac{du}{a^2 + u^2} = \frac{1}{a}\int\frac{dv}{1 + v^2}  		= \frac{1}{a}\,\arctan v + \hat{C}  		= \frac{1}{a}\,\arctan\left(\frac{u}{a}\right) + C \end{equation*}

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