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Evaluate a given integral equation

Define

    \[ F(x,a) = \int_0^x \frac{t^p}{(t^2+a^2)^q} \, dt, \]

with p,q positive integers and a > 0 a real number. Show that

    \[ F(x,a) = a^{p+1-2q} F \left( \frac{x}{a}, 1 \right). \]


Proof. Let

    \[ u = \frac{t}{a}, \qquad du = \frac{1}{a} \, dt. \]

Then,

    \begin{align*}  \int_0^x \frac{t^p}{(t^2+a^2)^q} \, dt &= a \int_{u(0)}^{u(x)} \frac{u^p a^p}{(u^2 a^2 + a^2)^q} \, du \\  &= a^{p+1} \int_{u(0)}^{u(x)} \frac{u^p}{a^{2q} (u^2+1)^q} \, du \\  &= a^{p+1-2q} \int_0^{\frac{x}{a}} \frac{u^p}{(u^2+1)^q} \, du \\  &= a^{p+1-2q} F \left( \frac{x}{a}, 1 \right). \qquad \blacksquare \end{align*}

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