Evaluate a given integral equation

by RoRi

October 18, 2015

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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Evaluate a given integral equation

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