Determine all continuous functions \(f:(0,\infty)\to(0,\infty)\) such that \[ \int_t^{t^3} f(x) \, dx = 2\int_1^t f(x)\,dx\] for all \(t>0\).
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Determine all continuous functions \(f:(0,\infty)\to(0,\infty)\) such that \[ \int_t^{t^3} f(x) \, dx = 2\int_1^t f(x)\,dx\] for all \(t>0\).