(1)

\begin{align*} \int f\left(\sqrt{a^{2}-x^{2}}\right)dx & =\int f\left(\sqrt{a^{2}-a^{2}\sin^{2}t}\right)d\left(a\sin t\right)\cnd{x=a\sin t}\\ & =a\int f\left(a\cos t\right)\cos tdt \end{align*}

(1)-2

\begin{align*} \int f\left(\sqrt{a^{2}-x^{2}}\right)dx & =\int f\left(\sqrt{a^{2}-a^{2}\tanh^{2}t}\right)d\left(a\tanh t\right)\cnd{x=a\tanh t}\\ & =a\int f\left(\frac{a}{\cosh t}\right)\frac{1}{\cosh^{2}t}dt \end{align*}

(2)

\begin{align*} \int f\left(\sqrt{a^{2}+x^{2}}\right)dx & =\int f\left(\sqrt{a^{2}+a^{2}\sinh^{2}t}\right)d\left(a\sinh t\right)\cnd{x=a\sinh t}\\ & =a\int f\left(a\cosh t\right)\cosh tdt \end{align*}

(2)-2

\begin{align*} \int f\left(\sqrt{a^{2}+x^{2}}\right)dx & =\int f\left(\sqrt{a^{2}+a^{2}\tan^{2}t}\right)d\left(a\tan t\right)\cnd{x=a\tan t}\\ & =a\int f\left(\frac{a}{\cos t}\right)\frac{1}{\cos^{2}t}dt \end{align*}

(2)-3

\begin{align*} \int f\left(\sqrt{a^{2}+x^{2}}\right)dx & =\int f\left(\left|\frac{t^{2}+a^{2}}{2t}\right|\right)d\left(\frac{a^{2}-t^{2}}{2t}\right)\cnd{t=x+\sqrt{x^{2}+a^{2}}}\\ & =-\frac{1}{2}\int f\left(\frac{t^{2}+a^{2}}{2t}\right)\frac{t^{2}+a^{2}}{t^{2}}dt \end{align*}

(3)

\begin{align*} \int f\left(\sqrt{x^{2}-a^{2}}\right)dx & =\int f\left(\sqrt{a^{2}\cosh^{2}t-a^{2}}\right)d\left(a\cosh t\right)\cnd{x=a\cosh t}\\ & =a\int f\left(a\sinh t\right)\sinh tdt \end{align*}

(3)-2

\begin{align*} \int f\left(\sqrt{x^{2}-a^{2}}\right)dx & =\int f\left(\sqrt{a^{2}\tanh^{-2}-a^{2}}\right)d\left(a\tanh^{-1}t\right)\cnd{x=a\tanh^{-1}t}\\ & =-a\int f\left(\frac{a}{\sinh t}\right)\frac{1}{\sinh^{2}t}dt \end{align*}

(3)-3

\begin{align*} \int f\left(\sqrt{x^{2}-a^{2}}\right)dx & =\int f\left(\left|\frac{t^{2}-a^{2}}{2t}\right|\right)d\left(\frac{t^{2}+a^{2}}{2t}\right)\cnd{t=x+\sqrt{x^{2}-a^{2}}}\\ & =\frac{1}{2}\int f\left(\frac{t^{2}-a^{2}}{2t}\right)\frac{t^{2}-a^{2}}{t}dt \end{align*}