Antiderivatives for Sums and Differences of Squares

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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Antiderivatives for Sums and Differences of Squares

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

Antiderivative of

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