If ∫ f( x )dx = ψ( x ) , then ∫ x ^ 5 f( x ^ 3 )dx , is equal to

$∫ {x}^{5}f\left({x}^{3}\right)dx$ $= ∫ {x}^{3} . {x}^{2}f\left({x}^{3}\right)dx$

Applying integration by parts, we get

$= {x}^{3} ∫ {x}^{2}f\left({x}^{3}\right)dx - ∫ \dfrac{d}{dx}\left({x}^{3}\right) ∫ {x}^{2}f\left({x}^{3}\right)dx$

$= \dfrac{1}{3}{x}^{3}ψ\left({x}^{3}\right) - ∫ {x}^{2}ψ\left({x}^{3}\right)dx + c$, where $c$ is the constant of integration.