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Prove an expression for the integral from 0 to 1 of (1 + x30) / (1+x60)

by RoRi

Prove that there exists a number c with 0 < c < 1 such that

    \[ \int_0^1 \frac{1+x^{30}}{1+x^{60}} \, dx = 1 + \frac{c}{31}. \]

Proof. For 0 < x < 1 we have

    \[ 1 < \frac{1+x^{30}}{1+x^{60}} < 1 + x^{30}. \]

(Since for 0 < x < 1 we know 1+x^{60} < 1+x^{30}.) Therefore, integrating the terms in the inequality from 0 to 1,

    \begin{align*} &\int_0^1 \, dx < \int_0^1 \frac{1+x^{30}}{1+x^{60}} \, dx < \int_0^1 (1+x^{30}) \, dt \\[9pt] \implies & 1 < \int_0^1 \frac{1+x^{30}}{1+x^{60}} \, dx < 1 + \frac{1}{31} \\[9pt] \implies & \int_0^1 \frac{1+x^{30}}{1+x^{60}} \, dx = 1 + \frac{c}{31} \qquad \text{for some } 0 < c < 1. \qquad \blacksquare \end{align*}

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